#6451763

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3884 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)(11,20)(12,17)(13,28)(14,19)(15,18)(16,22)(21,37)(23,31)(24,36)(25,30)(26,35)(27,33)(29,34)(32,38)(39,40);(9,11,15,24)(10,16,18,31)(12,25,27,39)(13,23,14,22)(17,32,33,37)(19,36,28,20)(21,29,38,26)(30,34,40,35);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,33)(19,34)(20,37)(22,25)(23,39)(24,38)(28,35)(31,40)(32,36);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,33)(19,34)(20,37)(22,25)(23,39)(24,38)(28,35)(31,40)(32,36);(10,18)(16,31)(17,33)(19,28)(20,36)(30,40)(32,37)(34,35)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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	#72318726
maximal normal subgroup of	#72318726

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358959);