#9224392

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

4160 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)(3,8)(4,6)(5,9)(7,11)(10,16)(12,14)(13,15)(18,19)(21,29)(23,25)(27,30)(33,34,39,41)(35,40,44,48)(36,45,37,38)(42,43,46,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,32,30,31)(33,35,39,44)(34,40,41,48)(36,46,37,42)(38,43,45,47);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,31,30,32)(33,36,39,37)(34,38,41,45)(35,42,44,46)(40,47,48,43);(1,4,6,2)(3,8,11,7)(5,9,14,12)(10,15,13,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,32,30,31)(33,37,39,36)(34,38,41,45)(35,42,44,46)(40,43,48,47);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,31)(27,29)(33,35)(34,40)(36,42)(37,46)(38,47)(39,44)(41,48)(43,45);(1,5,6,14)(2,9,4,12)(3,10,11,13)(7,15,8,16)(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,38)(34,37)(35,43)(36,41)(39,45)(40,42)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gi,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, 9131588);