#72328025

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

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

54655 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|b^2=gh,d^2=ghi,b^a=bg,c^b=cgi,d^a=di,d^b=di,d^c=dj,e^b=egh,e^d=egh,f^a=fgij,f^b=fgij,f^e=fgij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328045, #72328397
maximal normal subgroup of	#72328045, #72328397
maximal quotient of	#72335775

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, 2, F.7*F.8);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, 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);