#8873642

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=hi,e^2=ghi,f^2=gh,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, 8780838);