#624387

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

1704 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fghi,c^2=fi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

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, 531583);