#5958542

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2355 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5865738);