#10166141

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

4077 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=i,d^2=gh,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, 10073337);