#9160768

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

4139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067964);