#9206522

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

4079 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=gh,e^2=hi,f^2=h,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, 9113718);