#9717598

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

4509 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,13)(14,18)(15,23)(16,20)(17,24)(19,22)(26,32)(27,30)(28,34)(37,39);(3,8)(6,7)(9,11)(10,17)(12,22)(13,23)(14,15)(16,24)(18,19)(20,21)(25,26)(27,35)(28,29)(30,33)(31,32)(34,40)(36,37)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,16,14)(10,18,20,13)(11,22,24,15)(17,19,21,23)(25,27,31,30)(26,33,32,35)(28,38,34,36)(29,37,40,39);(1,2,5,4)(3,7,8,6)(9,13,16,18)(10,12,20,14)(11,23,24,19)(15,17,22,21)(25,28,31,34)(26,29,32,40)(27,36,30,38)(33,39,35,37);(1,3)(2,7)(4,6)(5,8)(9,14,16,12)(10,18,20,13)(11,22,24,15)(17,23,21,19)(25,29)(26,34)(27,37)(28,32)(30,39)(31,40)(33,36)(35,38);(1,4,5,2)(3,7,8,6)(9,15,16,22)(10,19,20,23)(11,14,24,12)(13,17,18,21)(25,30,31,27)(26,33,32,35)(28,38,34,36)(29,39,40,37)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=hi,f^2=h,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, 9624794);