#8917730

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

4410 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8824926);