#10143744

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

4181 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=hi,e^2=gh,f^2=h,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, 10050940);