#8688319

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

4417 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595515);