#10143427

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

4048 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)(17,18,23,25)(19,20,28,24)(21,26,32,30)(22,29,31,27)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(1,3,7,4)(2,8,6,9)(5,11,13,12)(10,15,14,16)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(33,35,39,45)(34,40,37,47)(36,44,46,38)(41,48,42,43);(18,25)(20,24)(26,30)(27,29)(33,36,39,46)(34,41,37,42)(35,38,45,44)(40,43,47,48);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,43)(47,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,37,39,34)(35,40,45,47)(36,42,46,41)(38,43,44,48);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(33,38,39,44)(34,43,37,48)(35,36,45,46)(40,41,47,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=ghi,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, 10050623);