#8227416

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

4226 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8134612);