#8688316

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

4545 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,20)(21,23)(22,24)(25,26,32,35)(27,30,31,33)(28,37,29,34)(36,40,39,38);(2,5)(6,7)(9,11)(10,13)(12,19)(14,17)(15,22)(16,18)(20,23)(21,24)(25,27,32,31)(26,33,35,30)(28,36,29,39)(34,40,37,38);(1,2)(3,6)(4,5)(7,8)(25,28,32,29)(26,34,35,37)(27,36,31,39)(30,38,33,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,21)(10,15,18,23)(11,19,17,24)(13,22,16,20)(25,29,32,28)(26,34,35,37)(27,36,31,39)(30,40,33,38);(9,11)(10,16)(12,19)(13,18)(14,17)(15,20)(21,24)(22,23)(25,30)(26,27)(28,38)(29,40)(31,35)(32,33)(34,36)(37,39);(9,13,14,16)(10,17,18,11)(12,20,21,22)(15,19,23,24)(25,31,32,27)(26,33,35,30)(28,39,29,36)(34,40,37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,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, 8595512);