#8712728

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

4598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=g,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, 8619924);