#6516515

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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:

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

Polycyclic presentation:

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

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, 6423711);