#8864564

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	#48658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

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

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, 8771760);