#748886

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=f,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 656082);