#9668380

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

4180 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)(3,10)(4,5)(6,7)(8,11)(9,16)(12,15)(13,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,34,39,43)(35,41,36,40)(37,42,48,38)(44,45,46,47);(2,7)(4,9)(8,10)(12,14)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,35,39,36)(34,40,43,41)(37,44,48,46)(38,45,42,47);(2,7)(3,11)(4,9)(13,15)(18,22)(19,26)(24,28)(27,32)(33,35,39,36)(34,41,43,40)(37,46,48,44)(38,45,42,47);(1,3)(2,8)(4,14)(5,15)(6,11)(7,10)(9,12)(13,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,36,39,35)(34,41,43,40)(37,44,48,46)(38,45,42,47);(1,4,6,9)(2,5,7,16)(3,12,11,14)(8,13,10,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,37)(34,42)(35,44)(36,46)(38,43)(39,48)(40,47)(41,45);(1,5)(2,9)(3,13)(4,7)(6,16)(8,14)(10,12)(11,15)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,38,39,42)(34,37,43,48)(35,45,36,47)(40,44,41,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,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, 9575576);