#6492535

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=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, 6399731);