#7658337

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

4334 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)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,21,24,22)(25,26)(27,34)(28,31)(29,36)(30,32)(33,38)(35,39)(37,40);(1,3,5,8)(2,6,4,7)(12,20)(16,23)(19,24)(21,22)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23);(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,23)(20,22)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565533);