#7516545

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	#61074	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1949 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,25)(13,16)(14,30)(15,20)(17,28)(18,23)(19,32)(22,39)(24,34)(26,38)(27,35)(29,37)(31,36)(33,40);(2,5)(6,8)(9,11,15,13)(10,16,20,21)(12,26,29,40)(14,23,32,24)(17,35,36,39)(18,30,34,19)(22,31,27,28)(25,33,37,38);(9,12,15,29)(10,17,20,36)(11,22,13,27)(14,31,32,28)(16,33,21,38)(18,35,34,39)(19,37,30,25)(23,26,24,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,18,20,34)(12,27,29,22)(14,24,32,23)(16,19,21,30)(17,35,36,39)(25,33,37,38)(26,31,40,28);(2,5)(6,8)(9,14,15,32)(10,19,20,30)(11,23,13,24)(12,28,29,31)(16,18,21,34)(17,25,36,37)(22,40,27,26)(33,39,38,35)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=i,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,g^b=gi,e^c=ei,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, 7423741);