#8720623

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	#47604	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627819);