#8633539

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

4001 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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, 8540735);