#9206540

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

4106 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)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,34)(35,42)(36,46)(37,40)(38,41)(39,45)(43,44)(47,48);(1,2,6,7)(3,4,10,9)(5,8,15,13)(11,12,16,14)(17,19)(18,22)(20,29)(21,30)(23,26)(24,25)(27,28)(31,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(2,7)(3,10)(8,13)(11,16)(17,20,23,30)(18,24,27,31)(19,29,26,21)(22,25,28,32)(33,36,38,40)(34,37,41,46)(35,43,45,47)(39,44,42,48);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,20)(18,24)(19,29)(21,26)(22,25)(23,30)(27,31)(28,32)(33,37,38,46)(34,40,41,36)(35,44,45,48)(39,47,42,43);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21,23,29)(18,25,27,32)(19,30,26,20)(22,24,28,31)(33,34)(35,42)(36,37)(38,41)(39,45)(40,46)(43,48)(44,47);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22,23,28)(18,26,27,19)(20,25,30,32)(21,31,29,24)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=h,e^2=ghi,f^2=gh,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, 9113736);