#6110940

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

1510 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=i,d^2=ghi,e^2=fgh,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, 6018136);