#6467395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3122	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2318 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)(13,14,17,21,18,22,32,39)(15,23,26,34,25,38,31,19)(16,27,30,43,29,20,35,37)(24,41,42,33,28,40,44,36);(2,3)(13,15)(14,19)(16,28)(17,31)(18,25)(20,36)(21,38)(22,34)(23,39)(24,29)(26,32)(27,33)(30,42)(35,44)(37,40)(41,43);(1,2,4,3)(6,9)(10,12)(13,16)(14,20)(15,24)(17,30)(18,29)(19,33)(21,37)(22,27)(23,40)(25,28)(26,42)(31,44)(32,35)(34,36)(38,41)(39,43);(1,3,4,2)(15,25)(19,34)(23,38)(24,28)(26,31)(33,36)(40,41)(42,44);(5,7,8,11)(6,10,9,12)(16,29)(20,27)(24,28)(30,35)(33,36)(37,43)(40,41)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374591);