#8665284

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,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, 8572480);