#864434

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1656 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=ghi,d^2=g,e^2=fgh,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, 771630);