#6631297

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631116
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,23,30,25)(20,27,21,26)(24,31,32,29);(2,5)(6,8)(11,15)(13,16)(17,19)(18,23)(20,31)(21,29)(22,30)(24,26)(25,28)(27,32);(3,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,28,32)(20,29,21,31)(23,27,25,26);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,19)(18,25)(20,31)(21,29)(22,30)(23,28)(24,27)(26,32);(17,20,22,21)(18,26,28,27)(19,29,30,31)(23,32,25,24)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,e^b=ei,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538493);