#8739246

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

3939 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(2,3)(7,9)(8,10)(15,16)(18,20)(24,27)(25,30)(26,31)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,18)(19,24,29,27)(21,30,28,25)(22,26,32,31)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=h,e^2=gh,f^2=gh,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, 8646442);