#10143324

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

4052 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)(3,8,13,15)(4,10,12,6)(5,11,14,16)(17,18,23,25)(19,20,28,24)(21,26,32,30)(22,29,31,27)(33,34,40,44)(35,41,46,48)(36,42,37,39)(38,43,45,47);(1,3)(2,8)(4,14)(5,12)(6,11)(7,13)(9,15)(10,16)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(33,35,40,46)(34,41,44,48)(36,47,37,43)(38,39,45,42);(2,9)(3,13)(4,12)(11,16)(18,25)(20,24)(26,30)(27,29)(33,36,40,37)(34,42,44,39)(35,43,46,47)(38,41,45,48);(1,4,7,12)(2,10,9,6)(3,14,13,5)(8,16,15,11)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,37,40,36)(34,39,44,42)(35,43,46,47)(38,41,45,48);(1,5,7,14)(2,11,9,16)(3,4,13,12)(6,8,10,15)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,38)(34,43)(35,39)(36,41)(37,48)(40,45)(42,46)(44,47);(1,6,7,10)(2,12,9,4)(3,11,13,16)(5,15,14,8)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(33,39)(34,36)(35,45)(37,44)(38,46)(40,42)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=i,e^2=h,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, 10050520);