#4062968

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=fh,e^2=gh,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, 3970164);