#3090216

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

1736 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 2997412);