#5956063

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 4194304
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

1642 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)(3,7,8,6)(10,13)(14,16)(17,18,21,20)(19,23,24,22)(27,33)(28,35)(30,36)(31,38);(3,8)(6,7)(9,10,12,13)(11,14,15,16)(27,33)(30,36)(34,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(17,18,21,20)(19,22,24,23)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=ghi,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5863259);