#7893279

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

4283 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)(9,10,15,12)(11,19,21,16)(13,22,24,17)(14,18,20,23)(25,26,31,34)(27,35,39,32)(28,29,38,40)(30,33,36,37);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,12,19)(13,20,24,14)(17,23,22,18)(25,27,31,39)(26,32,34,35)(28,30,38,36)(29,37,40,33);(1,2,5,4)(3,6,8,7)(10,12)(16,19)(17,22)(18,23)(25,28)(26,29)(27,36)(30,39)(31,38)(32,33)(34,40)(35,37);(9,12,15,10)(11,19,21,16)(13,22,24,17)(14,23,20,18)(25,29)(26,28)(27,37)(30,32)(31,40)(33,39)(34,38)(35,36);(9,13)(10,17)(11,20)(12,22)(14,21)(15,24)(16,23)(18,19)(25,28)(26,29)(27,30)(31,38)(32,37)(33,35)(34,40)(36,39);(1,4,5,2)(3,7,8,6)(9,14,15,20)(10,18,12,23)(11,13,21,24)(16,17,19,22)(25,30)(26,33)(27,38)(28,39)(29,32)(31,36)(34,37)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7800475);