#8466867

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,e^2=ghi,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, 8374063);