#7518587

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	#60433	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 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)(11,17)(12,25)(13,28)(14,20)(15,22)(16,21)(18,27)(19,29)(23,37)(24,31)(26,34)(30,35)(32,36)(33,38)(39,40);(2,5)(6,7)(9,11,16,25)(10,17,21,12)(13,23,30,39)(14,18,15,24)(19,26,36,33)(20,27,22,31)(28,37,35,40)(29,34,32,38);(9,12,16,17)(10,18,21,24)(11,22,25,20)(13,26,30,33)(14,27,15,31)(19,34,36,38)(23,28,39,35)(29,37,32,40);(9,13,16,30)(10,19,21,36)(11,23,25,39)(12,26,17,33)(14,29,15,32)(18,34,24,38)(20,35,22,28)(27,37,31,40);(1,3,4,8)(2,6,5,7)(9,14,16,15)(10,20,21,22)(11,24,25,18)(12,27,17,31)(13,29,30,32)(19,35,36,28)(23,38,39,34)(26,37,33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=fi,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, 7425783);