#8843851

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

4283 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8751047);