#857777

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 764973);