#8811227

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

4063 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)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,22,28,24)(20,29,31,25)(21,26,32,30)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(1,2,5,8)(3,6,10,9)(4,7,12,14)(11,13,16,15)(17,19,23,28)(18,24,27,22)(20,21,31,32)(25,26,29,30)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(3,10)(4,12)(6,9)(7,14)(17,20,23,31)(18,25,27,29)(19,21,28,32)(22,30,24,26)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,3,5,10)(2,6,8,9)(4,11,12,16)(7,13,14,15)(17,20)(18,25)(19,21)(22,30)(23,31)(24,26)(27,29)(28,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,21)(18,26)(19,20)(22,29)(23,32)(24,25)(27,30)(28,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,22,23,24)(18,19,27,28)(20,30,31,26)(21,29,32,25)(33,37,38,46)(34,40,41,48)(35,42,39,45)(36,43,44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8718423);