#7646797

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	#41007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4295 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)(11,20)(12,21)(14,16)(17,24)(25,26,32,30)(27,36,35,31)(28,33,40,38)(29,34,39,37);(1,2)(3,5)(4,6)(7,8)(9,10,15,19)(11,16,20,14)(12,17,21,24)(13,18,23,22)(25,27)(26,31)(28,34)(29,33)(30,36)(32,35)(37,40)(38,39);(9,11,15,20)(10,16,19,14)(12,13,21,23)(17,18,24,22)(25,28)(26,33)(27,37)(29,36)(30,38)(31,39)(32,40)(34,35);(9,12)(10,17)(11,13)(14,22)(15,21)(16,18)(19,24)(20,23)(25,29,32,39)(26,34,30,37)(27,33,35,38)(28,36,40,31);(9,13)(10,18)(11,21)(12,20)(14,17)(15,23)(16,24)(19,22)(25,30,32,26)(27,36,35,31)(28,38,40,33)(29,34,39,37);(1,3,4,7)(2,5,6,8)(9,14)(10,11)(12,22)(13,17)(15,16)(18,21)(19,20)(23,24)(25,31,32,36)(26,35,30,27)(28,39,40,29)(33,34,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,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, 7553993);