#6476584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(21,24)(25,30)(26,31)(27,29)(28,32);(9,11)(10,13)(12,15)(14,16)(17,18,24,31,22,26,21,23)(19,25,29,32,20,28,27,30);(2,5)(6,8)(11,15)(13,16)(17,19)(18,25)(20,22)(21,27)(23,30)(24,29)(26,28)(31,32);(11,15)(13,16)(17,20,22,19)(18,25,26,28)(21,29,24,27)(23,30,31,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,20)(25,28)(27,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6383780);