#9224336

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
prime radical	#49923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,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, 9131532);