#8739346

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

4094 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,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, 8646542);