#7655754

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

4462 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562950);