#3585830

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=ghi,d^2=fhi,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, 3493026);