#234480

order	512
nilpotency class	3
isoclinism	isoclinic class of #232215
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,23,26,36)(16,20,27,34)(17,29,31,21)(19,30,33,25)(24,37,38,44)(28,41,40,35)(32,42,43,39);(9,12)(13,15,18,26)(14,19,22,33)(16,24,27,38)(17,30,31,25)(20,32,34,43)(21,36,29,23)(28,42,40,39)(35,44,41,37);(2,4)(5,7)(6,9)(8,11)(10,12)(13,16)(14,20)(15,24)(17,28)(18,27)(19,32)(21,35)(22,34)(23,37)(25,39)(26,38)(29,41)(30,42)(31,40)(33,43)(36,44);(16,27)(20,34)(24,38)(28,40)(32,43)(35,41)(37,44)(39,42)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141676);