#6548100

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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)(17,18,21,27,22,28,23,29)(19,25,31,32,30,24,20,26);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,28,25)(20,21,31,23)(26,29,32,27);(10,13)(14,16)(17,20)(18,25)(19,23)(21,30)(22,31)(24,28)(26,27)(29,32);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,26)(19,21)(22,31)(23,30)(24,29)(25,27)(28,32);(11,15)(14,16)(17,21,22,23)(18,27,28,29)(19,20,30,31)(24,32,25,26)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,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, 6455296);