#8720653

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=hi,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, 8627849);