#6537632

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×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6444828);