#8324296

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231492);