#767113

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

1686 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)(10,13)(14,16)(18,22)(23,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,39)(38,40);(1,2,5,4)(3,6,7,8)(18,22)(23,24)(26,27)(28,35)(32,34)(36,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,3)(2,6)(4,8)(5,7)(17,19,21,20)(18,23,22,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

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, 674309);