#6811305

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#25390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1855 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718501);