#10168360

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3934 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18)(19,29)(20,26)(21,22)(23,27)(24,28)(25,31)(30,32)(33,34)(35,40)(36,43)(37,39)(38,45)(41,47)(42,44)(46,48);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19)(18,25)(20,23)(21,30)(22,27)(24,31)(26,32)(28,29)(33,35,39,43)(34,36,37,40)(38,42,47,46)(41,44,45,48);(2,9)(3,4)(10,14)(11,12)(17,20,24,21)(18,26,28,22)(19,23,31,30)(25,32,29,27)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,21,24,20)(18,26,28,22)(19,30,31,23)(25,32,29,27)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,22,24,26)(18,21,28,20)(19,27,31,32)(23,25,30,29)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,23,24,30)(18,27,28,32)(19,20,31,21)(22,29,26,25)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)

Polycyclic presentation:

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

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, 10075556);