#10143428

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

4066 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)(18,22)(19,25)(24,27)(26,28)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,39,45)(34,40,37,47)(36,44,46,38)(41,48,42,43);(2,9)(3,4)(10,16)(11,12)(17,19,21,25)(18,23,22,30)(20,26,29,28)(24,31,27,32)(33,36,39,46)(34,41,37,42)(35,38,45,44)(40,43,47,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,43)(47,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,37,39,34)(35,40,45,47)(36,42,46,41)(38,43,44,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,38,39,44)(34,43,37,48)(35,36,45,46)(40,41,47,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,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, 10050624);