#8994526

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

4229 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31)(34,36)(39,43)(40,44)(47,48);(2,7)(4,15)(8,11)(10,13)(17,19)(18,24)(20,29)(21,23)(22,26)(25,27)(28,30)(31,32)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(2,7)(3,12)(4,15)(5,16)(18,25)(19,21)(20,30)(22,32)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,30)(18,26,25,31)(19,29,21,28)(22,27,32,24)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,23,19)(18,27,25,24)(20,29,30,28)(22,31,32,26)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22)(18,28)(19,26)(20,27)(21,31)(23,32)(24,30)(25,29)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gi,e^2=h,f^2=g,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, 8901722);