#6467185

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,24,38)(16,20,28,36)(17,30,32,21)(19,31,33,26)(25,39,27,41)(29,44,43,37)(34,42,35,40);(2,3)(13,15)(14,19)(16,27)(17,31)(18,24)(20,35)(21,38)(22,33)(23,30)(25,28)(26,32)(29,40)(34,36)(37,39)(41,44)(42,43);(1,2,4,3)(6,9)(10,12)(15,24)(19,33)(23,38)(25,27)(26,31)(34,35)(39,41)(40,42);(1,3,4,2)(13,16,18,28)(14,20,22,36)(15,25,24,27)(17,29,32,43)(19,34,33,35)(21,37,30,44)(23,39,38,41)(26,40,31,42);(5,7,8,11)(6,10,9,12)(16,28)(20,36)(25,27)(29,43)(34,35)(37,44)(39,41)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374381);