#8726535

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633731);