#6535301

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18,23,28)(19,29,30,25)(20,31,21,26)(22,27,24,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,24)(21,22)(23,30)(26,32)(27,31)(28,29);(10,13)(14,16)(17,20,23,21)(18,25,28,29)(19,22,30,24)(26,32,31,27);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(17,21,23,20)(18,26,28,31)(19,22,30,24)(25,32,29,27);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,28,32)(19,20,30,21)(25,26,29,31)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6442497);