#9159896

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
prime radical	#43662	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=ghi,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, 9067092);