#7714593

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

4084 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)(13,21)(17,22)(18,24)(20,23)(25,26,32,29)(27,36,38,33)(28,39,31,34)(30,40,37,35);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,32,38)(26,33,29,36)(28,37,31,30)(34,35,39,40);(10,15)(12,16)(17,22)(20,23)(25,28,32,31)(26,34,29,39)(27,37,38,30)(33,35,36,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,29,32,26)(27,36,38,33)(28,34,31,39)(30,40,37,35);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,30)(26,35)(27,28)(29,40)(31,38)(32,37)(33,39)(34,36);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,31,32,28)(26,34,29,39)(27,30,38,37)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=i,f^2=g,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, 7621789);