#8718390

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

4398 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625586);