#7646277

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

4583 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:

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

Polycyclic presentation:

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

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, 7553473);