#3330663

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

1792 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)(3,7)(4,6)(5,8)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,40)(38,39);(1,3,6,5)(2,7,4,8)(17,18,20,22)(19,21,24,23)(26,27)(28,35)(29,37)(39,40);(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,39,38,40);(1,4,6,2)(3,7,5,8)(9,11,13,15)(10,14,12,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,16,15,14)(17,19)(18,21)(20,24)(22,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=ghi,d^2=fghi,e^2=fghi,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, 3237859);