#2969645

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2876841);