#4465683

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
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

1308 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=fh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4372879);