#9230857

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=gh,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, 9138053);