#8633573

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

4187 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)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19,23,22)(18,24,21,26)(20,29,30,27)(25,32,28,31)(33,34)(35,40)(36,37)(38,39)(41,45)(42,46)(43,48)(44,47);(5,14)(8,12)(10,13)(15,16)(18,21)(24,26)(25,28)(31,32)(34,39)(36,45)(40,47)(42,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32)(33,35)(34,40)(36,42)(37,46)(38,44)(39,47)(41,43)(45,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21,23,18)(19,24,22,26)(20,28,30,25)(27,31,29,32)(33,36)(34,41)(35,42)(37,39)(38,45)(40,43)(44,48)(46,47);(1,3)(2,9)(4,7)(5,10)(6,11)(8,16)(12,15)(13,14)(17,22,23,19)(18,24,21,26)(20,27,30,29)(25,32,28,31)(33,37,38,41)(34,36,39,45)(35,43,44,46)(40,48,47,42)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=g,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, 8540769);