#9720375

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

4028 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9627571);