#6438800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3243 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316094, #72316120, #72316163
maximal normal subgroup of	#72316094, #72316120, #72316163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345996);