#8665388

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4585 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=gh,e^2=g,f^2=h,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, 8572584);