#4397044

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1366 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=hi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4304240);