#927565

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

1622 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)(4,6)(7,8)(10,15)(11,13)(17,18,20,22)(19,21,23,24)(27,28)(30,31);(1,3,6,8)(2,7,4,5)(9,10,14,15)(11,12,13,16)(19,23)(21,24)(25,26,29,32)(27,31,28,30);(2,4)(3,8)(9,11,14,13)(10,16,15,12)(17,19,20,23)(18,21,22,24)(25,27,29,28)(26,30,32,31);(1,4,6,2)(3,7,8,5)(9,12,14,16)(10,11,15,13)(17,18)(19,21)(20,22)(23,24)(25,28,29,27)(26,31,32,30);(1,5,6,7)(2,8,4,3)(9,13,14,11)(10,16,15,12)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fi,d^2=g,e^2=f,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, 834761);