#9717743

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

4448 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,f^2=i,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, 9624939);