#7654745

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

4305 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)(9,10,15,19)(11,20,22,16)(12,17,23,13)(14,18,24,21)(25,26,32,30)(27,33,29,28)(31,34,39,38)(35,40,37,36);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,19,20)(12,21,23,18)(13,14,17,24)(25,27,32,29)(26,28,30,33)(31,35,39,37)(34,36,38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,18)(13,19)(14,16)(15,23)(20,24)(21,22)(25,28)(26,29)(27,30)(31,36)(32,33)(34,37)(35,38)(39,40);(1,2,5,4)(3,6,8,7)(9,13,15,17)(10,12,19,23)(11,14,22,24)(16,21,20,18)(25,29,32,27)(26,33,30,28)(31,37,39,35)(34,40,38,36);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,21)(13,19)(14,20)(15,23)(16,24)(18,22)(25,30,32,26)(27,33,29,28)(31,34,39,38)(35,36,37,40);(1,4,5,2)(3,7,8,6)(9,14)(10,18)(11,17)(12,16)(13,22)(15,24)(19,21)(20,23)(25,31,32,39)(26,34,30,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=g,e^2=gi,f^2=gi,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, 7561941);