#9706980

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9614176);