#8550903

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

3945 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:

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

Polycyclic presentation:

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

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, 8458099);