#8712257

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	C₂×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=gh,e^2=g,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, 8619453);