#7645928

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=h,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, 7553124);