#8466874

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
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4393 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=g,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, 8374070);