#3191596

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

1717 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fghi,d^2=fghi,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, 3098792);