#7642884

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	#39582	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550080);