#6452225

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318970, #72319082
maximal normal subgroup of	#72318970, #72319082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359421);