#2746414

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
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

1837 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)(11,15)(13,16)(17,18,21,22)(19,20,23,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(3,8)(5,7)(9,10,12,14)(11,13,15,16)(17,19,21,23)(18,20,22,24)(26,30)(27,34)(32,38)(35,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,36)(29,34)(30,33)(32,40)(35,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28,29,37)(26,32,30,38)(27,35,34,36)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 2653610);