#4457304

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1642 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=i,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

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, 4364500);