#7656982

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	#37616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 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)(9,10,15,20)(11,21,22,16)(12,18,23,17)(13,14,19,24)(25,26,31,29)(27,34,30,32)(28,36,38,33)(35,39,37,40);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,20,21)(12,19,23,13)(14,17,24,18)(25,27,31,30)(26,32,29,34)(28,35,38,37)(33,39,36,40);(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,24)(18,20)(25,26,31,29)(27,34,30,32)(28,33,38,36)(35,40,37,39);(9,13,15,19)(10,14,20,24)(11,12,22,23)(16,17,21,18)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,19)(13,22)(14,16)(15,23)(17,20)(21,24)(25,29,31,26)(27,34,30,32)(28,36,38,33)(35,40,37,39);(1,4,5,2)(3,7,8,6)(9,14)(10,19)(11,17)(12,21)(13,20)(15,24)(16,23)(18,22)(25,30,31,27)(26,32,29,34)(28,37,38,35)(33,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564178);