#8534616

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

4286 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,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^c=egi,f^d=fgi

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, 8441812);