#6807089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×#3198	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714285);