#2244473

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1438 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fghi,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2151669);