#7518283

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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

1878 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)(11,21)(12,24)(13,16)(14,19)(15,31)(17,28)(18,29)(20,30)(22,35)(23,34)(25,33)(26,36)(27,38)(32,37)(39,40);(2,5)(6,7)(9,11,14,13)(10,16,19,21)(12,25,27,40)(15,29,30,23)(17,35,37,36)(18,31,34,20)(22,28,26,32)(24,39,38,33);(9,12)(10,17)(11,22)(13,26)(14,27)(15,32)(16,33)(18,36)(19,37)(20,38)(21,39)(23,25)(24,31)(28,30)(29,40)(34,35);(9,13,14,11)(10,18,19,34)(12,26,27,22)(15,23,30,29)(16,20,21,31)(17,36,37,35)(24,33,38,39)(25,28,40,32);(1,3,4,8)(2,6,5,7)(10,19)(16,21)(17,37)(18,34)(20,31)(24,38)(33,39)(35,36)

Polycyclic presentation:

2^9|b^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=cgi,d^b=dh,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, 7425479);