#8533423

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

4026 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440619);