#7655244

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

4312 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)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26,31,30)(27,34,37,32)(28,29,33,39)(35,36,38,40);(1,3,5,8)(2,6,4,7)(25,27)(26,32)(28,38)(29,36)(30,34)(31,37)(33,35)(39,40);(1,2,5,4)(3,6,8,7)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,28)(26,29)(27,35)(30,39)(31,33)(32,40)(34,36)(37,38);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29,31,39)(26,33,30,28)(27,36,37,40)(32,35,34,38);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,30,31,26)(27,34,37,32)(28,39,33,29)(35,36,38,40);(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,27)(26,34)(28,35)(29,36)(30,32)(31,37)(33,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,e^2=gh,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, 7562440);