#8943762

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	#42982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8850958);