#6673289

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,19,20)(16,27,30,35)(17,31,33,22)(21,28,37,25)(26,34,32,38)(29,42,43,40)(36,39,44,41);(1,2)(3,4)(6,9)(7,11)(13,15,18,19)(14,20,23,24)(16,28,30,25)(17,32,33,26)(21,35,37,27)(22,38,31,34)(29,39,43,41)(36,40,44,42);(2,4)(13,16,18,30)(14,21,23,37)(15,25,19,28)(17,29,33,43)(20,27,24,35)(22,36,31,44)(26,39,32,41)(34,40,38,42);(6,9)(10,12)(13,17)(14,22)(15,26)(16,29)(18,33)(19,32)(20,34)(21,36)(23,31)(24,38)(25,39)(27,40)(28,41)(30,43)(35,42)(37,44);(2,4)(5,7,8,11)(6,10,9,12)(16,30)(17,33)(21,37)(22,31)(25,28)(26,32)(27,35)(34,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6580485);