#7643441

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

4345 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)(9,10,14,12)(11,16,19,15)(13,20,21,17)(18,22,24,23)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(1,2,4,5)(3,6,7,8)(9,11,14,19)(10,15,12,16)(13,18,21,24)(17,22,20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,28,30,38)(26,32,34,36)(27,35,29,37)(31,39,33,40);(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^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, 7550637);