#7658421

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

4404 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)(9,10,15,18)(11,17,21,14)(12,22,13,16)(19,23,20,24)(25,26,31,34)(27,35,37,32)(28,29,38,40)(30,33,39,36);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,14,18,17)(12,19,13,20)(16,23,22,24)(25,27,31,37)(26,32,34,35)(28,36,38,33)(29,39,40,30);(9,12,15,13)(10,16,18,22)(11,19,21,20)(14,23,17,24)(25,28)(26,29)(27,33)(30,32)(31,38)(34,40)(35,39)(36,37);(9,11)(10,17)(12,19)(13,20)(14,18)(15,21)(16,24)(22,23)(25,29)(26,28)(27,30)(31,40)(32,33)(34,38)(35,36)(37,39);(1,2,5,4)(3,6,8,7)(9,13,15,12)(10,16,18,22)(11,20,21,19)(14,23,17,24)(25,28)(26,29)(27,36)(30,35)(31,38)(32,39)(33,37)(34,40);(1,4,5,2)(3,7,8,6)(9,14,15,17)(10,11,18,21)(12,23,13,24)(16,19,22,20)(25,30)(26,33)(27,29)(28,32)(31,39)(34,36)(35,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=gi,f^2=gi,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, 7565617);