#7643544

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	#35189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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)(12,19)(16,21)(17,23)(22,24)(25,26)(27,35)(28,29)(30,32)(31,33)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,31,30)(26,32,33,35)(28,36,39,38)(29,37,34,40);(10,14)(15,20)(16,21)(22,24)(26,33)(29,34)(32,35)(37,40);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,34)(27,36)(29,33)(30,38)(31,39)(32,40)(35,37);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,31,34)(26,28,33,39)(27,37,30,40)(32,36,35,38);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,30,31,27)(26,35,33,32)(28,38,39,36)(29,40,34,37)

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=g,f^2=hi,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, 7550740);