#8393350

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300546);