#7643514

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	#35522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=h,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, 7550710);