#7643543

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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)(12,19)(16,21)(17,23)(22,24)(25,26)(27,31)(28,37)(29,39)(30,32)(33,38)(34,35)(36,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,30,36)(26,31,32,40)(28,35,38,29)(33,39,37,34);(1,4,5,2)(3,7,8,6)(10,14)(15,20)(16,21)(22,24)(26,32)(31,40)(33,37)(34,39);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,39)(32,37)(34,40);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,29,30,35)(26,34,32,39)(27,28,36,38)(31,33,40,37)

Polycyclic presentation:

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

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, 7550739);