#8421834

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	#57964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4578 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=g,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, 8329030);