#7647276

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	#42971	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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)(14,22)(17,21)(19,20)(23,24)(25,26)(27,34)(28,37)(29,32)(30,38)(31,33)(35,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10,15,13)(11,18,12,16)(14,17,22,21)(19,24,20,23);(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27,31,29)(26,32,33,34)(28,35,38,39)(30,36,37,40);(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,29,31,27)(26,32,33,34)(28,35,38,39)(30,40,37,36);(1,3,4,7)(2,5,6,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7554472);