#7654190

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

4455 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)(11,14,19,16)(12,18,13,17)(21,24,22,23)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,27,31,30)(26,32,35,36)(28,37,34,39)(29,38,33,40);(9,12,15,13)(10,17,20,18)(11,21,19,22)(14,23,16,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,38)(39,40);(1,4,5,2)(3,7,8,6)(9,13,15,12)(10,18,20,17)(11,22,19,21)(14,24,16,23)(25,28,31,34)(26,33,35,29)(27,37,30,39)(32,40,36,38);(9,11)(10,14)(12,21)(13,22)(15,19)(16,20)(17,23)(18,24)(25,29,31,33)(26,34,35,28)(27,38,30,40)(32,39,36,37);(9,14)(10,19)(11,20)(12,23)(13,24)(15,16)(17,22)(18,21)(25,30,31,27)(26,32,35,36)(28,39,34,37)(29,38,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,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, 7561386);