#9224296

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

4200 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)(17,18,22,26)(19,27,28,23)(20,24,31,30)(21,32,29,25)(33,34,39,37)(35,43,45,40)(36,46,44,41)(38,42,48,47);(1,3)(2,4)(5,9)(6,14)(7,11)(8,13)(10,15)(12,16)(17,19,22,28)(18,23,26,27)(20,29,31,21)(24,32,30,25)(33,35,39,45)(34,40,37,43)(36,38,44,48)(41,42,46,47);(1,4,7,5)(2,3,9,11)(6,12,15,13)(8,10,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31)(33,34,39,37)(35,40,45,43)(36,41,44,46)(38,42,48,47);(1,5,7,4)(2,3,9,11)(6,12,15,13)(8,14,16,10)(17,18)(19,23)(20,30)(21,32)(22,26)(24,31)(25,29)(27,28)(33,36)(34,41)(35,38)(37,46)(39,44)(40,42)(43,47)(45,48);(1,6)(2,8)(3,10)(4,12)(5,13)(7,15)(9,16)(11,14)(17,20,22,31)(18,24,26,30)(19,21,28,29)(23,25,27,32)(33,37,39,34)(35,43,45,40)(36,41,44,46)(38,42,48,47);(1,3)(2,5)(4,9)(6,14)(7,11)(8,12)(10,15)(13,16)(17,21)(18,25)(19,20)(22,29)(23,24)(26,32)(27,30)(28,31)(33,38)(34,42)(35,44)(36,45)(37,47)(39,48)(40,46)(41,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=i,e^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, 9131492);