#9668328

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

4613 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575524);