#8726483

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

4147 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,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, 8633679);