#503270

order	512
nilpotency class	4
isoclinism	isoclinic class of #503268
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25779
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320162, #72343955, #72343960
maximal normal subgroup of	#72320162, #72343955, #72343960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410466);