#10147130

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

4144 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,e^2=i,f^2=g,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, 10054326);