#9224323

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

4165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,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, 9131519);