#3397233

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
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

1626 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)(11,14)(12,16)(13,15)(17,18,21,20)(19,22,24,23)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,18,21,20)(19,23,24,22)(26,30)(31,33)(32,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,36)(29,30)(32,38)(33,35)(34,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,20)(19,22,24,23)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,21,18)(19,23,24,22)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fh,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3304429);