#3553032

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1844 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=ghi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3460228);