#950260

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 857456);