#7642930

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

4336 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)(9,10,15,12)(11,19,20,16)(13,17,23,21)(14,22,24,18)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,18)(14,21)(15,20)(17,24)(22,23)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(11,20)(13,23)(16,19)(17,21)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(1,3,4,7)(2,5,6,8)(9,12,15,10)(11,19,20,16)(13,21,23,17)(14,22,24,18)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(9,13,15,23)(10,17,12,21)(11,18,20,22)(14,16,24,19)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39);(9,14)(10,18)(11,21)(12,22)(13,16)(15,24)(17,20)(19,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^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, 7550126);