#7654896

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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)(10,16)(11,12)(18,24)(19,20);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,17)(14,18)(15,16)(19,21)(20,22)(23,24)(26,30)(31,37)(32,38)(39,40);(9,11,15,12)(10,13,16,17)(14,19,23,20)(18,21,24,22)(25,26,29,30)(27,31,34,37)(28,32,36,38)(33,39,35,40);(9,12,15,11)(10,17,16,13)(14,20,23,19)(18,22,24,21);(1,3,4,7)(2,6,5,8)(9,13,15,17)(10,11,16,12)(14,22,23,21)(18,20,24,19)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562092);