#8211622

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

4169 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)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(34,40)(35,36)(37,48)(44,47);(2,4)(7,9)(8,12)(14,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,34)(35,41)(36,45)(37,38)(39,40)(42,48)(43,47)(44,46);(5,13)(8,12)(10,16)(14,15)(19,25)(23,26)(27,29)(31,32)(33,35,39,36)(34,41,40,45)(37,43,48,46)(38,47,42,44);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,36,39,35)(34,41,40,45)(37,46,48,43)(38,47,42,44);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,37)(34,42)(35,43)(36,46)(38,40)(39,48)(41,44)(45,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,38,39,42)(34,37,40,48)(35,44,36,47)(41,46,45,43)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=hi,e^2=gh,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, 8118818);