#6110983

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

1514 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)(3,6)(4,5)(7,8)(10,11)(15,16)(17,18)(19,22)(20,21)(23,24)(27,33)(28,35)(30,36)(31,38);(2,5)(6,8)(9,10,14,11)(12,16,13,15)(17,19,21,24)(18,22,20,23)(27,33)(30,36)(34,39)(37,40);(3,7)(6,8)(9,11,14,10)(12,16,13,15)(17,19)(18,23)(20,22)(21,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,13)(10,15,11,16)(17,20,21,18)(19,23,24,22)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,12)(10,16,11,15)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=ghi,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, 6018179);