#72327970

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 7311 conjugacy classes of subgroups; only the 4237 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(2,4)(5,6,7,8)(9,10,13,14)(11,12,16,15)(17,18,22,24)(19,23,21,20);(1,2,3,4)(5,6,7,8)(9,11,13,16)(10,12,14,15);(1,2,3,4)(5,6,7,8)(9,10,13,14)(11,15,16,12)(17,19,22,21)(18,23,24,20);(2,4)(5,6,7,8)(10,14)(11,16)(17,20)(18,21)(19,24)(22,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,24)(22,23);(9,12,13,15)(10,11,14,16)(17,21,22,19)(18,23,24,20)

Polycyclic presentation:

2^10|a^2=h,b^2=ghj,c^2=gh,d^2=ghij,f^2=gij,b^a=bg,c^a=cg,c^b=cgi,d^a=dgij,d^b=di,d^c=dij,e^a=eghj,e^b=eghj,e^c=eghj,e^d=eghj,f^a=fgij,f^e=fgij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328000, #72336936
maximal normal subgroup of	#72328000, #72336936
maximal quotient of	#72336109

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.7*F.8*F.10);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);