#6500526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407722);