#7520557

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	#30555	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1727 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,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, 7427753);