#9015531

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

4092 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922727);