#8718407

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

4107 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625603);