#7643422

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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)(11,15)(12,21)(13,23)(14,18)(16,22)(17,19)(20,24)(25,26,30,33)(27,31,35,28)(29,37,38,32)(34,36,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,18,24)(12,19,22,13)(16,23,21,17)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(9,10,14,18)(11,15,20,24)(12,16,22,21)(13,17,19,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(9,13,14,19)(10,17,18,23)(11,12,20,22)(15,16,24,21)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=g,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, 7550618);