#8230693

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

4055 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(33,34,40,44)(35,45,47,41)(36,38,37,42)(39,43,48,46);(2,7)(4,8)(9,14)(12,15)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,35,40,47)(34,41,44,45)(36,46,37,43)(38,48,42,39);(5,13)(9,14)(10,16)(12,15)(18,23)(21,30)(24,27)(28,32)(33,36,40,37)(34,42,44,38)(35,46,47,43)(39,45,48,41);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,37,40,36)(34,38,44,42)(35,46,47,43)(39,45,48,41);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,38)(34,36)(35,39)(37,44)(40,42)(41,43)(45,46)(47,48);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,39)(34,43)(35,38)(36,41)(37,45)(40,48)(42,47)(44,46)

Polycyclic presentation:

2^9|a^2=h,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, 8137889);