#9184714

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

4253 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)(19,24)(20,23)(21,25)(22,28)(26,31)(27,29)(30,32)(33,34)(35,41)(36,42)(37,46)(38,39)(40,45)(43,44)(47,48);(1,3,7,5)(2,8,6,10)(4,11,12,14)(9,15,13,16)(17,19,23,25)(18,24,20,21)(22,27,31,32)(26,30,28,29)(34,39)(36,46)(40,41)(43,47);(4,12)(9,13)(11,14)(15,16)(18,20)(19,25)(26,28)(27,32)(33,35,38,45)(34,40,39,41)(36,43,46,47)(37,44,42,48);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20,23,18)(19,21,25,24)(22,28,31,26)(27,29,32,30)(33,35)(34,41)(36,47)(37,44)(38,45)(39,40)(42,48)(43,46);(1,5,7,3)(2,8,6,10)(4,11,12,14)(9,16,13,15)(17,21)(18,25)(19,20)(22,30)(23,24)(26,27)(28,32)(29,31)(33,36,38,46)(34,42,39,37)(35,43,45,47)(40,48,41,44);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37,38,42)(34,36,39,46)(35,44,45,48)(40,43,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,e^2=g,f^2=g,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, 9091910);