#929712

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

1786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(5,8)(11,15)(14,16)(17,18,20,22)(19,21,24,23)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,39)(38,40);(10,13)(14,16)(17,18,20,22)(19,23,24,21)(26,27)(28,35)(32,34)(36,40);(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fi,d^2=h,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, 836908);