#8393347

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

4172 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,28)(22,24)(23,27)(25,31)(26,32)(29,30)(33,34)(35,39)(36,41)(37,45)(38,40)(42,48)(43,44)(46,47);(2,6)(3,10)(8,13)(11,16)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,35,38,41)(34,39,40,36)(37,46,48,43)(42,44,45,47);(4,12)(8,13)(11,16)(14,15)(18,24)(21,23)(25,28)(30,32)(34,40)(35,41)(42,45)(43,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,36,38,39)(34,41,40,35)(37,47,48,44)(42,43,45,46);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20,22,31)(18,25,24,28)(19,26,27,29)(21,30,23,32)(33,37,38,48)(34,42,40,45)(35,43,41,46)(36,44,39,47);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,21,22,23)(18,19,24,27)(20,30,31,32)(25,26,28,29)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|b^2=i,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, 8300543);