#8679938

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4264 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587134);