#9214731

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,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, 9121927);