#9206575

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

4122 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)(3,8)(4,12)(5,7)(6,11)(9,14)(10,15)(13,16)(17,18,23,22)(19,26,21,24)(20,25,28,29)(27,32,30,31)(33,34)(35,40)(36,39)(37,41)(38,44)(42,47)(43,45)(46,48);(1,3,5,11)(2,6,7,8)(4,10,14,13)(9,15,12,16)(17,19,23,21)(18,24,22,26)(20,27,28,30)(25,31,29,32)(33,35,39,41)(34,40,36,37)(38,43,47,48)(42,46,44,45);(2,7)(6,8)(9,12)(15,16)(20,28)(25,29)(27,30)(31,32)(34,36)(35,41)(42,44)(43,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,15)(12,14)(13,16)(17,20,23,28)(18,25,22,29)(19,27,21,30)(24,31,26,32)(33,36,39,34)(35,37,41,40)(38,44,47,42)(43,45,48,46);(1,3)(2,8)(4,13)(5,11)(6,7)(9,15)(10,14)(12,16)(17,21,23,19)(18,24,22,26)(20,27,28,30)(25,32,29,31)(33,37)(34,41)(35,36)(38,46)(39,40)(42,43)(44,48)(45,47);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,22,23,18)(19,26,21,24)(20,29,28,25)(27,32,30,31)(33,38,39,47)(34,42,36,44)(35,43,41,48)(37,45,40,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=gh,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, 9113771);