#7658614

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

4303 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)(12,20)(15,21)(18,24)(22,23)(25,26,32,35)(27,36,31,30)(28,37,40,33)(29,34,38,39);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(25,27,32,31)(26,30,35,36)(28,34,40,39)(29,33,38,37);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,28)(26,33)(27,34)(29,36)(30,38)(31,39)(32,40)(35,37);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,23)(17,21)(19,24)(25,29,32,38)(26,34,35,39)(27,37,31,33)(28,36,40,30);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,30,32,36)(26,27,35,31)(28,38,40,29)(33,34,37,39);(1,4,5,3)(2,8,6,7)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,31,32,27)(26,30,35,36)(28,39,40,34)(29,33,38,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=g,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, 7565810);