#8227441

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

4266 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,15)(14,16)(33,34)(35,41)(36,46)(37,42)(38,43)(39,40)(44,45)(47,48);(2,7)(3,10)(4,13)(14,15)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(34,40)(37,43)(41,46)(44,47);(2,7)(5,8)(9,12)(14,15)(18,23)(21,30)(24,27)(28,32)(33,35,39,36)(34,41,40,46)(37,44,43,47)(38,48,42,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36,39,35)(34,41,40,46)(37,47,43,44)(38,48,42,45);(1,4,6,13)(2,9,7,12)(3,11,10,16)(5,14,8,15)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,37,39,43)(34,42,40,38)(35,44,36,47)(41,45,46,48);(1,5,6,8)(2,3,7,10)(4,14,13,15)(9,11,12,16)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,38,39,42)(34,43,40,37)(35,45,36,48)(41,44,46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=ghi,e^2=gi,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, 8134637);