#7654121

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

4466 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,e^2=i,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, 7561317);