#7657913

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	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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)(9,10,16,20)(11,21,22,17)(12,23,14,18)(13,15,19,24)(25,26,30,31)(27,34,32,28)(29,33,37,38)(35,40,39,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,13)(14,19)(15,18)(16,22)(20,21)(23,24)(26,31)(27,32)(33,38)(35,39);(9,12,16,14)(10,18,20,23)(11,13,22,19)(15,21,24,17)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(9,13,16,19)(10,15,20,24)(11,12,22,14)(17,18,21,23)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(1,3,4,7)(2,5,6,8)(9,14,16,12)(10,18,20,23)(11,19,22,13)(15,21,24,17);(9,15)(10,19)(11,18)(12,21)(13,20)(14,17)(16,24)(22,23)(25,29,30,37)(26,33,31,38)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565109);