#7651295

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	#42477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,38)(26,32,36,37)(28,39,34,30)(29,40,33,35);(10,14)(15,17)(16,22)(23,24)(27,38)(30,39)(32,37)(35,40);(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,28)(26,33)(27,39)(29,36)(30,38)(31,34)(32,35)(37,40);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29,31,33)(26,34,36,28)(27,40,38,35)(30,37,39,32);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,30,31,39)(26,35,36,40)(27,28,38,34)(29,37,33,32)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7558491);