#9668376

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

4054 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34,39,36)(35,40,45,43)(37,41,44,46)(38,42,48,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,42,46,47);(2,6)(3,10)(8,12)(11,16)(18,25)(19,20)(26,32)(27,28)(33,34,39,36)(35,43,45,40)(37,46,44,41)(38,42,48,47);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36,39,34)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37,39,44)(34,41,36,46)(35,38,45,48)(40,42,43,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,46)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575572);