#8943951

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

4258 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8851147);