#9206567

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

4108 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:

(1,2,7,6)(3,10,5,8)(4,9,12,13)(11,16,14,15)(17,18)(19,26)(20,25)(21,31)(22,24)(23,29)(27,32)(28,30)(33,34)(35,40)(36,39)(37,41)(38,44)(42,47)(43,45)(46,48);(1,3,7,5)(2,8,6,10)(4,11,12,14)(9,15,13,16)(17,19,22,29)(18,23,24,26)(20,27,30,31)(21,28,32,25)(33,35,39,41)(34,40,36,37)(38,43,47,48)(42,46,44,45);(4,12)(9,13)(11,14)(15,16)(18,24)(20,30)(23,26)(27,31)(34,36)(35,41)(42,44)(43,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,26,32,23)(33,36,39,34)(35,37,41,40)(38,44,47,42)(43,45,48,46);(1,5,7,3)(2,8,6,10)(4,11,12,14)(9,16,13,15)(17,19)(18,26)(20,31)(21,25)(22,29)(23,24)(27,30)(28,32)(33,37)(34,41)(35,36)(38,46)(39,40)(42,43)(44,48)(45,47);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,27)(19,28)(20,26)(22,32)(23,30)(24,31)(25,29)(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=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, 9113763);