#72233707

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
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68179 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=j,d^2=j,e^2=gh,f^2=h,b^a=bhi,c^b=cj,d^a=dj,d^c=dij,e^a=eg,e^b=ehij,f^b=fhij,f^e=fg

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233717, #72234554, C₂×#72233707
maximal normal subgroup of	#72233717, #72234554, C₂×#72233707
maximal quotient of	#72334419, C₂×#72233707, #72335021, …

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