#6549218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

1427 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,25,23,20,21,29)(19,24,27,32,28,26,30,31);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,23,28)(18,24,20,26)(21,30,22,27)(25,32,29,31);(10,11)(14,15)(17,20)(18,23)(19,26)(21,29)(22,25)(24,28)(27,32)(30,31);(1,4,5,3)(2,6,7,8)(12,16)(14,15);(9,12,13,16)(10,14,11,15)(17,21,23,22)(18,25,20,29)(19,27,28,30)(24,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6456414);