#7645842

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

4357 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553038);