#8701689

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4194 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)(9,10,15,18)(11,12,13,16)(14,17,23,24)(19,20,21,22);(3,7)(5,8)(9,11,15,13)(10,16,18,12)(14,19,23,21)(17,22,24,20)(26,31)(27,34)(33,40)(35,37);(1,2)(3,5)(4,6)(7,8)(25,26,30,31)(27,32,34,28)(29,33,39,40)(35,38,37,36);(1,3)(2,5)(4,7)(6,8)(9,12,15,16)(10,11,18,13)(14,22,23,20)(17,21,24,19)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,16,18,12)(14,19,23,21)(17,20,24,22)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,37,40,35);(9,14,15,23)(10,17,18,24)(11,19,13,21)(12,20,16,22)(25,29,30,39)(26,33,31,40)(27,35,34,37)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608885);