#4057269

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1827 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=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, 3964465);