#6618358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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)(13,14,19,22)(15,20,25,35)(16,28,30,37)(17,23,18,31)(21,32,36,29)(24,38,39,43)(26,34,27,41)(33,42,44,40);(1,2,4,3)(6,10)(8,11)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,33)(22,35)(23,34)(28,38)(29,40)(30,39)(31,41)(32,42)(36,44)(37,43);(1,3,4,2)(5,6)(7,8)(9,10)(11,12)(13,16,19,30)(14,21,22,36)(15,24,25,39)(17,29,18,32)(20,33,35,44)(23,37,31,28)(26,40,27,42)(34,43,41,38);(5,7,9,12)(6,11,10,8)(14,22)(15,25)(21,36)(23,31)(24,39)(26,27)(28,37)(40,42);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,23,22,31)(15,26,25,27)(16,29,30,32)(20,34,35,41)(21,37,36,28)(24,40,39,42)(33,43,44,38)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6525554);