#2952655

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

1660 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)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,23)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(1,3,4,8)(2,5,7,6)(9,11,13,12)(10,14,16,15)(17,19)(18,20)(21,22)(23,24)(27,31)(28,34)(35,39)(36,40);(9,11,13,12)(10,15,16,14)(17,20)(18,22)(19,23)(21,24)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,15,16,14)(17,20)(18,19)(21,24)(22,23)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,23)(25,29,30,38)(26,32,33,37)(27,35,31,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=ghi,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2859851);