#6534189

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(3,7)(4,8)(5,6)(17,18)(19,26)(20,24)(21,25)(22,30)(23,28)(27,31)(29,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,29,23)(20,30,22,24)(26,28,32,31);(10,13)(14,16)(17,19,21,29)(18,23,25,27)(20,28,22,31)(24,32,30,26);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,30)(21,25)(22,24)(26,28)(27,29)(31,32);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,20,21,22)(18,24,25,30)(19,28,29,31)(23,32,27,26)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6441385);