#10166177

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

4045 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10073373);