#8821951

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
prime radical	#47991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8729147);