#9224368

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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,e^2=g,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, 9131564);