#8230694

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

4049 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:

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

Polycyclic presentation:

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

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, 8137890);