#7656923

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

4393 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,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, 7564119);