#8826210

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

4168 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,e^2=gh,f^2=g,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, 8733406);