#9163984

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(17,18,23,21)(19,24,27,29)(20,25,31,30)(22,26,32,28)(33,34,38,43)(35,39,46,40)(36,41,42,37)(44,45,47,48);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,19,23,27)(18,24,21,29)(20,28,31,26)(22,30,32,25)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(4,13)(9,12)(11,14)(15,16)(19,27)(20,31)(24,29)(25,30)(33,35,38,46)(34,40,43,39)(36,44,42,47)(37,45,41,48);(1,4,7,13)(2,9,6,12)(3,11,5,14)(8,15,10,16)(17,20)(18,25)(19,28)(21,30)(22,24)(23,31)(26,27)(29,32)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(1,5,7,3)(2,8,6,10)(4,11,13,14)(9,16,12,15)(17,21,23,18)(19,29,27,24)(20,25,31,30)(22,26,32,28)(33,36)(34,41)(35,44)(37,43)(38,42)(39,45)(40,48)(46,47);(1,6,7,2)(3,10,5,8)(4,12,13,9)(11,16,14,15)(17,22)(18,26)(19,30)(20,24)(21,28)(23,32)(25,27)(29,31)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,44,40,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=g,e^2=ghi,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, 9071180);