#7654031

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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 7561227);