#6516700

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(19,28)(20,26)(21,27)(22,24)(23,31)(25,29)(30,32);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,24)(25,32)(26,28)(27,31);(10,13)(11,15)(17,19)(18,25)(20,22)(21,29)(23,30)(24,32)(26,27)(28,31);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,26,24,28)(21,29,23,30)(25,31,32,27);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(17,21,22,23)(18,27,24,31)(19,29,20,30)(25,26,32,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6423896);