#8422100

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

4472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8329296);