#10143767

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

4232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,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, 10050963);