#8688315

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

4337 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)(3,5)(4,7)(6,8)(12,20)(15,23)(19,24)(21,22)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(2,3)(6,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,21)(20,23)(22,24)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,19)(14,17)(15,22)(16,18)(20,24)(21,23);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,15,18,23)(11,19,17,24)(13,22,16,21)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,21)(20,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,13,14,16)(10,17,18,11)(12,21,20,22)(15,19,23,24)(25,29,30,26)(27,33,35,31)(28,32,36,37)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,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, 8595511);