#8037571

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4148 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)(12,20)(15,22)(18,24)(21,23)(25,26,31,35)(27,36,39,32)(28,40,34,30)(29,33,38,37);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,26,31,35)(27,36,39,32)(28,30,34,40)(29,37,38,33);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,39)(26,32,35,36)(28,33,34,37)(29,30,38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,28,31,34)(26,30,35,40)(27,37,39,33)(29,36,38,32);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29)(26,33)(27,30)(28,36)(31,38)(32,34)(35,37)(39,40);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,30)(26,34)(27,38)(28,35)(29,39)(31,40)(32,37)(33,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7944767);