#2667017

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1734 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=fi,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2574213);