#8701706

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×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608902);