#8806867

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

4050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714063);