#4456678

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
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

1394 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=gi,d^2=fh,e^2=fg,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, 4363874);