#8813947

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
prime radical	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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)(3,5)(4,8)(6,10)(7,12)(9,14)(11,15)(13,16)(17,18)(19,28)(20,31)(21,22)(23,24)(25,29)(26,30)(27,32)(34,36)(35,38)(41,44)(42,45);(1,3,6,12)(2,7,10,5)(4,11,14,16)(8,13,9,15)(18,24)(19,29)(21,27)(26,31)(33,34,39,36)(35,40,38,43)(37,44,46,41)(42,48,45,47);(1,4,6,14)(2,8,10,9)(3,11,12,16)(5,15,7,13)(17,19)(18,25)(20,27)(21,30)(22,31)(23,29)(24,28)(26,32)(33,35,39,38)(34,40,36,43)(37,45,46,42)(41,48,44,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,15)(8,10)(12,16)(17,20,23,30)(18,26,24,31)(19,27,29,21)(22,25,32,28)(33,36,39,34)(35,40,38,43)(37,44,46,41)(42,47,45,48);(1,5)(2,3)(4,13)(6,7)(8,16)(9,11)(10,12)(14,15)(17,21,23,27)(18,22,24,32)(19,30,29,20)(25,31,28,26)(33,37,39,46)(34,41,36,44)(35,42,38,45)(40,47,43,48);(1,3)(2,5)(4,11)(6,12)(7,10)(8,15)(9,13)(14,16)(17,22)(18,27)(19,31)(20,25)(21,24)(23,32)(26,29)(28,30)(33,38,39,35)(34,40,36,43)(37,45,46,42)(41,47,44,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=hi,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, 8721143);