#6487736

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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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)(5,6)(7,10)(8,11)(9,12)(13,14)(15,24)(16,22)(17,23)(18,20)(19,31)(21,26)(25,39)(27,36)(28,37)(29,34)(30,38)(32,33)(35,42)(40,43)(41,44);(1,2,3,4)(7,8)(10,11)(14,20)(21,33)(22,34)(23,31)(24,36)(35,43)(37,38)(39,44);(5,7,9,8)(6,10,12,11)(13,15,18,27)(14,21,20,33)(16,25,29,41)(17,26,19,32)(22,35,34,43)(23,36,31,24)(28,40,30,42)(37,39,38,44);(5,8,9,7)(6,11,12,10)(13,16)(14,22)(15,25)(17,30)(18,29)(19,28)(20,34)(21,35)(23,38)(24,39)(26,42)(27,41)(31,37)(32,40)(33,43)(36,44);(6,12)(10,11)(13,17,18,19)(14,23,20,31)(15,26,27,32)(16,28,29,30)(21,36,33,24)(22,37,34,38)(25,40,41,42)(35,39,43,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=h,e^2=i,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, 6394932);