#8926611

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

4200 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833807);