#9065531

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

4164 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)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26,31,35)(27,32,30,29)(28,34,39,33)(36,38,37,40);(1,2)(3,4)(5,6)(7,8)(10,14)(15,17)(16,20)(23,24)(25,27,31,30)(26,32,35,29)(28,37,39,36)(33,38,34,40);(2,6)(3,7)(10,14)(15,17)(16,20)(23,24)(25,28,31,39)(26,33,35,34)(27,36,30,37)(29,38,32,40);(3,7)(4,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,29)(26,27)(28,38)(30,35)(31,32)(33,36)(34,37)(39,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,28)(26,34)(27,36)(29,40)(30,37)(31,39)(32,38)(33,35);(1,4)(2,3)(5,8)(6,7)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,30,31,27)(26,32,35,29)(28,37,39,36)(33,40,34,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972727);