#8726584

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=ghi,f^2=gi,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, 8633780);