#7643183

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	#39795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 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)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,31,28)(27,35,37,32)(29,33,36,38)(30,39,40,34);(1,2)(3,5)(4,6)(7,8)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(10,14)(15,17)(16,20)(22,23)(25,26,31,28)(27,35,37,32)(29,38,36,33)(30,34,40,39);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,31,26)(27,35,37,32)(29,38,36,33)(30,39,40,34);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,31,36)(26,33,28,38)(27,30,37,40)(32,34,35,39);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=i,f^2=h,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, 7550379);