#8898554

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

4225 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805750);