#751595

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

1732 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)(25,26)(27,28)(29,34)(30,31)(32,33)(35,36)(37,38)(39,40);(9,11,13,12)(10,14,16,15)(18,22)(23,24)(26,31)(27,33)(29,37)(36,39);(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,20)(18,23,22,24)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20,21,19)(18,24,22,23)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,d^2=h,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, 658791);