#6494207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6401403);