#9224338

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	#49589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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)(3,11)(4,9)(5,14)(6,7)(8,12)(10,16)(13,15)(17,18)(19,24)(20,30)(21,22)(23,26)(25,29)(27,32)(28,31)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,42,48,44);(2,7)(4,13)(8,11)(10,14)(17,19)(18,24)(20,31)(21,27)(22,32)(23,29)(25,26)(28,30)(33,35)(34,40)(36,44)(37,43)(38,41)(39,45)(42,47)(46,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,20,23,21)(18,22,26,30)(19,27,29,31)(24,28,25,32)(33,34,39,37)(35,40,45,43)(36,41,47,46)(38,42,48,44);(1,3)(2,8)(4,14)(5,15)(6,12)(7,11)(9,16)(10,13)(17,21,23,20)(18,22,26,30)(19,27,29,31)(24,32,25,28)(33,36)(34,41)(35,44)(37,46)(38,40)(39,47)(42,45)(43,48);(1,4,6,13)(2,9,7,15)(3,10,12,14)(5,11,16,8)(17,22)(18,21)(19,28)(20,26)(23,30)(24,31)(25,27)(29,32)(33,37,39,34)(35,43,45,40)(36,41,47,46)(38,42,48,44);(1,5)(2,10)(3,9)(4,8)(6,16)(7,14)(11,13)(12,15)(17,19)(18,25)(20,27)(21,31)(22,32)(23,29)(24,26)(28,30)(33,38)(34,42)(35,41)(36,43)(37,44)(39,48)(40,47)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,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, 9131534);