#8700445

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

4145 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)(9,10,15,19)(11,20,14,16)(12,22,23,17)(13,18,24,21)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(2,5)(6,7)(9,11,15,14)(10,16,19,20)(12,18,23,21)(13,17,24,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,17,19,22)(11,21,14,18)(13,20,24,16)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,40)(36,39);(1,3,4,8)(2,6,5,7)(9,13)(10,18)(11,17)(12,20)(14,22)(15,24)(16,23)(19,21)(25,28,30,27)(26,32,34,31)(29,35,38,36)(33,39,37,40);(9,14,15,11)(10,16,19,20)(12,21,23,18)(13,17,24,22)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=h,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, 8607641);