#8739230

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

4081 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)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34,39,41)(35,43,37,40)(36,42,47,46)(38,44,48,45);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,35,39,37)(34,40,41,43)(36,44,47,45)(38,46,48,42);(2,3)(7,9)(8,10)(15,16)(20,28)(23,31)(27,32)(29,30)(34,41)(36,47)(40,43)(44,45);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,23)(19,27)(21,30)(22,28)(24,29)(25,32)(26,31)(33,36,39,47)(34,42,41,46)(35,45,37,44)(38,40,48,43);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,37,39,35)(34,43,41,40)(36,44,47,45)(38,46,48,42);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,38)(34,44)(35,46)(36,43)(37,42)(39,48)(40,47)(41,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=h,e^2=h,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, 8646426);