#6630694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629009
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,21)(22,26)(23,25)(24,31)(28,29)(30,32);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,31)(18,24,26,27)(20,28,25,32)(21,30,23,29);(1,3,5,4)(2,7,6,8)(17,20,21,18,22,25,23,26)(19,28,30,24,31,32,29,27);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,22,23)(18,25,26,20)(19,29,31,30)(24,28,27,32);(17,21,22,23)(18,20,26,25)(19,30,31,29)(24,28,27,32)

Polycyclic presentation:

2^9|b^2=h,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537890);