#8755290

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

4121 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662486);