#10143751

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

4043 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18)(19,27)(20,30)(21,26)(22,24)(23,25)(28,32)(29,31)(33,34,39,38)(35,42,45,40)(36,41,37,46)(43,48,44,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,23,22)(18,24,25,27)(20,28,26,31)(21,32,30,29)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,47)(46,48);(2,6)(7,9)(8,12)(15,16)(18,25)(20,26)(24,27)(28,31)(33,36,39,37)(34,41,38,46)(35,43,45,44)(40,47,42,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20)(18,21)(19,28)(22,31)(23,26)(24,32)(25,30)(27,29)(33,37,39,36)(34,41,38,46)(35,44,45,43)(40,47,42,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,26)(19,29)(20,25)(22,32)(23,30)(24,28)(27,31)(33,38,39,34)(35,42,45,40)(36,46,37,41)(43,48,44,47);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22,23,19)(18,27,25,24)(20,31,26,28)(21,29,30,32)(33,35)(34,42)(36,43)(37,44)(38,40)(39,45)(41,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=ghi,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, 10050947);