#8873796

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

4131 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780992);