#6506416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1426 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)(3,8)(4,6)(5,7)(17,18,24,31,22,26,23,27)(19,28,21,32,29,25,30,20);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,24)(22,29)(23,30)(26,28)(27,32);(9,11)(10,13)(12,16)(14,15)(17,20,22,32)(18,21,26,30)(19,27,29,31)(23,28,24,25);(11,16)(13,15)(17,21)(18,20)(19,23)(22,30)(24,29)(25,27)(26,32)(28,31);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(19,29)(21,30)(27,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=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, 6413612);