#7659043

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	#37541	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 7566239);