#8898553

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

4211 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)(19,29)(20,26)(21,22)(23,27)(24,28)(25,32)(30,31)(33,34)(35,44)(36,47)(37,48)(38,42)(39,40)(41,46)(43,45);(1,3,6,11)(2,7,9,16)(4,13,5,10)(8,15,12,14)(17,19)(18,25)(20,30)(21,27)(22,31)(23,26)(24,32)(28,29)(33,35,40,46)(34,41,39,44)(36,45,38,37)(42,43,47,48);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,14,16,15)(17,20,24,21)(18,26,28,22)(19,30,32,27)(23,29,31,25)(33,36,40,38)(34,42,39,47)(35,45,46,37)(41,43,44,48);(1,5,6,4)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,21,24,20)(18,26,28,22)(19,27,32,30)(23,29,31,25)(33,37)(34,43)(35,36)(38,46)(39,48)(40,45)(41,47)(42,44);(1,3)(2,7)(4,10)(5,13)(6,11)(8,14)(9,16)(12,15)(17,22,24,26)(18,21,28,20)(19,31,32,23)(25,27,29,30)(33,38,40,36)(34,42,39,47)(35,45,46,37)(41,48,44,43);(1,2)(3,7)(4,12)(5,8)(6,9)(10,15)(11,16)(13,14)(17,23,24,31)(18,27,28,30)(19,26,32,22)(20,25,21,29)(33,39,40,34)(35,44,46,41)(36,42,38,47)(37,48,45,43)

Polycyclic presentation:

2^9|a^2=i,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, 8805749);