#8720624

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

4186 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18)(19,27)(20,22)(21,24)(23,25)(26,31)(28,29)(30,32)(33,34,39,43)(35,40,45,47)(36,46,38,41)(37,42,48,44);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,29,26,32)(33,35,39,45)(34,40,43,47)(36,42,38,44)(37,46,48,41);(4,13)(8,12)(11,14)(15,16)(18,25)(20,31)(24,27)(28,30)(33,36,39,38)(34,41,43,46)(35,44,45,42)(37,40,48,47);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,20,23,31)(18,26,25,22)(19,28,21,30)(24,29,27,32)(33,37,39,48)(34,42,43,44)(35,41,45,46)(36,47,38,40);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,29,26,32)(33,34)(35,40)(36,41)(37,42)(38,46)(39,43)(44,48)(45,47);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,22,23,26)(18,20,25,31)(19,29,21,32)(24,30,27,28)(33,38,39,36)(34,41,43,46)(35,44,45,42)(37,47,48,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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, 8627820);