#8460760

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

4568 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8367956);