#8734923

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

4031 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^2=h,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, 8642119);