#8679874

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

4204 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587070);