#8688358

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

4611 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)(4,6)(5,8)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,22)(23,24)(25,26,30,34)(27,31,36,33)(28,37,29,32)(35,40,38,39);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,19,20)(13,21,18,23)(14,22,17,24)(25,27)(26,31)(28,38)(29,35)(30,36)(32,40)(33,34)(37,39);(1,2)(3,5)(4,6)(7,8)(25,28,30,29)(26,32,34,37)(27,35,36,38)(31,39,33,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,19,20)(13,23,18,21)(14,22,17,24)(25,26)(27,33)(28,32)(29,37)(30,34)(31,36)(35,40)(38,39);(9,13)(10,17)(11,21)(12,23)(14,19)(15,18)(16,24)(20,22)(25,27)(26,33)(28,35)(29,38)(30,36)(31,34)(32,40)(37,39);(9,14,15,17)(10,18,19,13)(11,22,12,24)(16,23,20,21)(25,29,30,28)(26,32,34,37)(27,35,36,38)(31,40,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,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, 8595554);