#6608367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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:

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

Polycyclic presentation:

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

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, 6515563);