#7714587

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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:

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

Polycyclic presentation:

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

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, 7621783);