#3161013

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1671 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)(10,14)(12,15)(19,23)(21,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,2,4,6)(3,5,8,7)(10,14)(11,16)(17,18,20,22)(19,21,23,24)(26,30)(27,34)(28,36)(38,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,15,14,12)(17,19,20,23)(18,21,22,24)(25,26)(27,33)(28,37)(29,30)(31,34)(32,36)(35,38)(39,40);(1,3,4,8)(2,5,6,7)(9,12,13,15)(10,16,14,11)(25,28,29,36)(26,32,30,37)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=fgi,e^2=g,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, 3068209);