#7714594

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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)(11,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24)(25,26)(27,34)(28,29)(30,38)(31,33)(32,39)(35,36)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27)(26,28)(29,31)(30,35)(32,36)(33,34)(37,39)(38,40);(11,18)(14,15)(19,24)(22,23)(25,28,31,34)(26,27,33,29)(30,36,39,40)(32,35,38,37);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,29,31,27)(26,28,33,34)(30,37,39,35)(32,36,38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,28)(26,27)(29,33)(30,36)(31,34)(32,35)(37,38)(39,40);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,17,20)(11,19,18,24)(14,22,15,23)(25,30,31,39)(26,32,33,38)(27,35,29,37)(28,36,34,40)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=gi,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, 7621790);