#9141665

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

4165 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9048861);