#8679981

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4370 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)(10,15)(16,17)(18,21)(23,24)(25,26,31,29)(27,32,30,34)(28,37,38,33)(35,40,36,39);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,24)(20,23)(25,29,31,26)(27,32,30,34)(28,37,38,33)(35,39,36,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,28)(26,33)(27,36)(29,37)(30,35)(31,38)(32,40)(34,39);(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,30,31,27)(26,34,29,32)(28,36,38,35)(33,40,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587177);