#2626851

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(9,10,13,11)(12,15,16,14)(18,22)(20,27)(23,28)(26,31)(33,34,38,39)(35,42,43,40)(36,37,46,41)(44,47,48,45);(1,2,4,6)(3,5,7,8)(9,11,13,10)(12,15,16,14)(18,22)(19,25)(24,29)(26,31)(34,39)(35,43)(36,46)(45,47);(1,2)(3,5)(4,6)(7,8)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,43)(34,40,39,42)(36,44,46,48)(37,45,41,47);(9,12)(10,14)(11,15)(13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,46)(34,41,39,37)(35,44,43,48)(40,47,42,45);(1,3,4,7)(2,5,6,8)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,41)(34,36,39,46)(35,45,43,47)(40,44,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2534047);