#8118861

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

4078 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)(3,9)(4,6)(5,12)(7,11)(8,13)(10,14)(15,16)(21,29)(24,31)(27,28)(30,32)(33,34)(35,37)(36,41)(38,48)(39,42)(40,45)(43,47)(44,46);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,35)(34,40)(36,43)(37,42)(38,44)(39,45)(41,46)(47,48);(2,4)(7,9)(8,12)(14,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,36,39,38)(34,41,42,48)(35,43,45,44)(37,47,40,46);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,35)(34,40)(36,43)(37,42)(38,44)(39,45)(41,46)(47,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,37,39,40)(34,35,42,45)(36,46,38,47)(41,44,48,43);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,36)(34,41,42,48)(35,44,45,43)(37,47,40,46)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

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, 8026057);