#9232792

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

4188 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=hi,e^2=gh,f^2=ghi,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, 9139988);