#7642630

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

4445 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=g,f^2=h,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, 7549826);