#7653426

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,31,34)(27,35,30,32)(28,29,33,39)(36,40,38,37);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,31,30)(26,32,34,35)(28,36,33,38)(29,37,39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,26,31,34)(27,35,30,32)(28,29,33,39)(36,40,38,37);(1,2,5,4)(3,6,8,7)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28,31,33)(26,29,34,39)(27,36,30,38)(32,37,35,40);(1,4,5,2)(3,7,8,6)(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,29)(26,33)(27,37)(28,34)(30,40)(31,39)(32,38)(35,36);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,37,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=ghi,f^2=hi,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, 7560622);