#9206547

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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 9113743);