#7564440

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	#61203	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1797 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=g,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,g^b=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471636);