#880591

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=fhi,e^2=f,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, 787787);