#734147

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

1668 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=gi,d^2=gi,e^2=gh,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, 641343);