#7653407

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 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)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,36)(28,29)(30,34)(31,33)(32,35)(37,38)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,32,31)(26,33,35,36)(28,39,30,37)(29,38,34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,28,32,30)(26,29,35,34)(27,37,31,39)(33,40,36,38);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,29,32,34)(26,28,35,30)(27,38,31,40)(33,39,36,37);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,30,32,28)(26,34,35,29)(27,37,31,39)(33,40,36,38);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,31,32,27)(26,33,35,36)(28,39,30,37)(29,40,34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7560603);