#8359977

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

4093 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)(3,9,11,7)(5,8,13,12)(10,15,16,14)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(34,37)(35,36)(42,45)(43,44);(2,4)(7,9)(8,12)(14,15)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(5,13)(8,12)(10,16)(14,15)(19,26)(22,24)(27,29)(30,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,4,6,2)(3,7,11,9)(5,12,13,8)(10,14,16,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8267173);