#7646526

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
prime radical	#35293	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,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, 7553722);