#8602496

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,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,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509692);