#9720435

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

4480 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=gi,f^2=gh,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, 9627631);