#3231227

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

1630 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fgh,e^2=ghi,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, 3138423);