#3254920

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

1844 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=f,e^2=fhi,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, 3162116);