#8127636

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

4072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8034832);