#950608

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

1566 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fg,d^2=gi,e^2=fh,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, 857804);