#7658942

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

4302 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566138);