#9224295

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

4191 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 9131491);