#8741966

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

4106 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(2,7)(3,8)(9,16)(10,13)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(34,40)(35,36)(42,47)(43,44);(2,7)(3,8)(9,16)(10,13)(20,28)(25,30)(27,29)(31,32)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8649162);