#5993044

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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

1376 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5900240);