#9029616

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
prime radical	#40866	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8936812);