#8227413

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

4235 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(34,40)(35,36)(37,48)(44,47);(2,6)(7,9)(8,13)(14,15)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,34)(35,41)(36,45)(37,38)(39,40)(42,48)(43,47)(44,46);(4,12)(8,13)(10,16)(14,15)(18,23)(21,30)(24,27)(28,32)(33,35,39,36)(34,41,40,45)(37,43,48,46)(38,47,42,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36,39,35)(34,41,40,45)(37,46,48,43)(38,47,42,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,37)(34,42)(35,43)(36,46)(38,40)(39,48)(41,44)(45,47);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,38,39,42)(34,37,40,48)(35,44,36,47)(41,46,45,43)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=i,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

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, 8134609);