#8230657

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

4065 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:

(1,2,6,5)(3,9,11,7)(4,12,14,8)(10,15,13,16)(33,34)(35,41)(36,44)(37,38)(39,40)(42,46)(43,45)(47,48);(1,3,6,11)(2,7,5,9)(4,13,14,10)(8,16,12,15)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(34,40)(37,47)(41,44)(42,45);(2,5)(7,9)(8,12)(15,16)(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)(4,10)(5,9)(6,11)(8,15)(12,16)(13,14)(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);(1,4,6,14)(2,8,5,12)(3,10,11,13)(7,15,9,16)(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);(1,5,6,2)(3,9,11,7)(4,12,14,8)(10,16,13,15)(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=h,b^2=hi,c^2=gh,d^2=ghi,e^2=hi,f^2=h,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, 8137853);