#9203898

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

4252 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)(4,9)(5,14)(6,7)(10,16)(11,15)(12,13)(17,18,22,20)(19,23,26,24)(21,25,29,28)(27,30,32,31)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47);(2,7)(4,13)(8,15)(10,14)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32)(33,35)(34,38)(36,45)(37,46)(39,42)(40,41)(43,44)(47,48);(2,7)(3,11)(9,12)(10,14)(19,26)(23,24)(27,32)(30,31)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,41,44,48);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,36)(34,40)(35,45)(37,42)(38,41)(39,46)(43,47)(44,48);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,16,15)(17,20,22,18)(19,24,26,23)(21,25,29,28)(27,30,32,31)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,40,44,47);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,39,44)(34,42,43,35)(36,41,46,48)(37,47,45,40)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9111094);