#8550904

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 8458100);