#8664681

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

4218 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)(11,20)(12,17)(13,23)(14,18)(15,19)(16,22)(21,24)(25,26,32,29)(27,31,34,39)(28,30,38,33)(35,40,37,36);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,14)(13,21)(15,22)(17,24)(18,23)(19,20)(25,27,32,34)(26,31,29,39)(28,37,38,35)(30,36,33,40);(1,3,5,6)(2,4,7,8)(9,12,15,13)(10,17,19,23)(11,14,22,21)(16,24,20,18)(25,28,32,38)(26,33,29,30)(27,35,34,37)(31,36,39,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,12)(10,17,19,23)(11,21,22,14)(16,24,20,18)(25,29,32,26)(27,31,34,39)(28,30,38,33)(35,36,37,40);(1,2)(3,4)(5,7)(6,8)(9,14,15,21)(10,18,19,24)(11,12,22,13)(16,23,20,17)(25,30)(26,28)(27,36)(29,38)(31,37)(32,33)(34,40)(35,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,16)(12,17)(13,23)(14,24)(15,19)(18,21)(20,22)(25,31)(26,34)(27,29)(28,36)(30,35)(32,39)(33,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gh,e^2=g,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, 8571877);