#3168314

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

1592 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=fgh,d^2=fghi,e^2=gh,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, 3075510);