#6131433

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1512 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)(3,5)(4,6)(7,8)(10,13)(14,15)(17,18,20,22)(19,23,24,21)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,15,16,14)(19,24)(21,23)(26,29)(31,33)(32,36)(38,39);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,14,13,15)(17,18)(19,23)(20,22)(21,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgh,c^2=fhi,d^2=hi,e^2=gh,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, 6038629);