#8284456

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

4233 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8191652);