#8158728

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	#41166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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)(5,10)(7,12)(8,13)(11,14)(15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(34,37)(35,36)(42,45)(43,44);(2,3)(7,9)(8,10)(14,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,3)(7,9)(8,10)(14,15)(20,29)(24,28)(26,32)(30,31)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8065924);