#8678977

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

4503 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=h,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, 8586173);