#8633808

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

4197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541004);