#7652335

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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559531);