#3580417

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

1392 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fgi,d^2=gi,e^2=fg,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, 3487613);