#7518584

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60429	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7425780);