#3827288

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

1723 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)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,24)(21,23)(25,26,30,31)(27,29,34,32)(28,33,37,39)(35,38,40,36);(1,2,4,6)(3,5,8,7)(17,19,21,24)(18,22,23,20)(26,31)(27,34)(33,39)(35,40);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,16,15)(17,20)(18,19)(21,22)(23,24)(25,27,30,34)(26,32,31,29)(28,35,37,40)(33,36,39,38);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,22)(23,24)(25,29,30,32)(26,27,31,34)(28,36,37,38)(33,40,39,35)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=gi,d^2=fgi,e^2=gi,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, 3734484);