#506361

order	512
nilpotency class	4
isoclinism	isoclinic class of #500851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28225
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

577 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)(4,18)(5,23)(6,26)(7,11)(8,15)(9,21)(10,20)(12,24)(14,37)(16,33)(17,44)(19,45)(22,52)(25,53)(27,48)(28,43)(29,55)(30,57)(31,49)(32,58)(34,40)(35,39)(36,42)(38,56)(41,54)(46,51)(47,50)(59,62)(60,64)(61,63);(1,3,5,15,24,42,48,57,7,16,25,43,6,17,20,39)(2,8,27,33,26,35,23,36,11,28,10,13,12,30,53,44)(4,14,21,40,50,55,58,64,19,38,49,62,22,41,51,63)(9,29,45,59,46,37,47,60,31,54,18,34,32,56,52,61);(4,19)(8,28)(9,31)(13,33)(14,38)(18,45)(21,49)(22,50)(30,35)(32,46)(36,44)(40,62)(41,55)(47,52)(51,58)(63,64);(1,4,20,51,6,22,25,49,7,19,48,58,24,50,5,21)(2,9,23,47,12,32,27,45,11,31,53,52,26,46,10,18)(3,14,39,63,17,41,43,62,16,38,57,64,42,55,15,40)(8,29,36,60,30,56,33,59,28,54,44,61,35,37,13,34)

Polycyclic presentation:

2^9|b^2=eg,d^2=eh,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,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217290, #72224998, #72225006
maximal normal subgroup of	#72217290, #72224998, #72225006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413557);