#8701641

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

4153 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)(25,26,31,30)(27,32,35,37)(28,33,36,39)(29,34,38,40);(2,5)(6,7)(9,11,15,14)(10,16,19,20)(12,18,23,21)(13,17,24,22)(27,35)(28,36)(32,37)(33,39);(1,2)(3,6)(4,5)(7,8)(25,27,31,35)(26,32,30,37)(28,29,36,38)(33,34,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,17,19,22)(11,21,14,18)(13,20,24,16)(25,28,31,36)(26,33,30,39)(27,29,35,38)(32,34,37,40);(1,3,4,8)(2,6,5,7)(9,13)(10,18)(11,17)(12,20)(14,22)(15,24)(16,23)(19,21)(25,29)(26,34)(27,36)(28,35)(30,40)(31,38)(32,39)(33,37);(9,14,15,11)(10,16,19,20)(12,21,23,18)(13,17,24,22)(25,30,31,26)(27,37,35,32)(28,33,36,39)(29,34,38,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gi,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, 8608837);