#9221174

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9128370);