#8873793

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

4135 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18,23,20)(19,21,25,26)(22,28,31,24)(27,32,30,29)(33,34)(35,41)(36,43)(37,45)(38,40)(39,47)(42,44)(46,48);(2,6)(7,9)(8,12)(15,16)(17,19)(18,21)(20,26)(22,30)(23,25)(24,32)(27,31)(28,29)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(3,10)(7,9)(11,13)(15,16)(18,20)(19,25)(24,28)(27,30)(33,34,38,40)(35,36,43,41)(37,39,47,45)(42,44,46,48);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,20,23,18)(19,26,25,21)(22,28,31,24)(27,32,30,29)(33,36,38,41)(34,35,40,43)(37,44,47,48)(39,42,45,46);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,21,23,26)(18,19,20,25)(22,29,31,32)(24,27,28,30)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8780989);