#8701696

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

4232 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)(3,5)(4,7)(6,8)(10,14)(15,22)(16,19)(23,24);(2,3)(6,7)(9,10,13,14)(11,15,18,22)(12,16,21,19)(17,23,20,24)(26,31)(27,34)(33,40)(35,37);(1,2,5,3)(4,6,8,7)(25,26,30,31)(27,32,34,28)(29,33,39,40)(35,38,37,36);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,37,40,35);(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,29,30,39)(26,33,31,40)(27,35,34,37)(28,36,32,38)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=h,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, 8608892);