#504006

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,25)(12,32)(13,22)(14,21)(15,23)(16,40)(17,24)(18,28)(19,29)(20,33)(26,41)(27,47)(30,48)(31,53)(34,62)(35,49)(36,66)(37,63)(38,46)(39,43)(42,65)(44,60)(45,54)(50,64)(51,55)(52,56)(57,61)(58,69)(59,68)(67,72)(70,71);(2,5)(6,7)(9,11,14,29,16,30,26,54,17,31,38,60,15,39,13,28)(10,18,22,43,23,44,46,53,24,45,41,48,40,19,21,25)(12,27,34,57,36,58,55,71,37,59,64,72,35,65,33,56)(20,42,49,67,50,68,63,70,51,69,66,61,62,47,32,52);(9,11,26,54,15,39,14,29,17,31,13,28,16,30,38,60)(10,19,46,43,40,45,22,25,24,44,21,48,23,18,41,53)(12,27,55,71,35,65,34,57,37,59,33,56,36,58,64,72)(20,47,63,67,62,69,49,52,51,68,32,61,50,42,66,70);(1,3,4,8)(2,6,5,7)(9,12,14,34,16,36,26,55,17,37,38,64,15,35,13,33)(10,20,22,49,23,50,46,63,24,51,41,66,40,62,21,32)(11,27,29,57,30,58,54,71,31,59,60,72,39,65,28,56)(18,42,43,67,44,68,53,70,45,69,48,61,19,47,25,52)

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=eh,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,g^b=gi,g^c=gi

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, 411202);