#8688393

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

4378 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,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, 8595589);