#8602788

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

4183 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8509984);