#456589

order	512
nilpotency class	3
isoclinism	isoclinic class of #456588
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16284
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1356 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)(3,15,19,42)(4,10)(5,25,28,11)(6,12)(7,13)(8,14)(9,18,33,30)(16,34,46,43)(17,44)(20,52,50,31)(21,35,54,63)(22,36)(23,37)(24,38)(26,39,55,57)(27,41,29,40)(32,48)(45,60)(47,64)(49,61)(51,56)(53,59)(58,62);(1,3,6,30,8,19,7,18)(2,9,12,42,14,33,13,15)(4,20,22,16,24,50,23,46)(5,17,29,48,28,49,27,51)(10,34,36,31,38,43,37,52)(11,32,41,61,25,56,40,44)(21,53,55,45,54,47,26,58)(35,62,57,59,63,60,39,64);(1,4,8,24)(2,10,14,38)(3,16,19,46)(5,26,28,55)(6,23,7,22)(9,31,33,52)(11,39,25,57)(12,37,13,36)(15,43,42,34)(17,47,49,53)(18,50,30,20)(21,29,54,27)(32,60,56,62)(35,41,63,40)(44,64,61,59)(45,51,58,48);(1,5)(2,11)(3,17)(4,21)(6,29)(7,27)(8,28)(9,32)(10,35)(12,41)(13,40)(14,25)(15,44)(16,45)(18,51)(19,49)(20,53)(22,55)(23,26)(24,54)(30,48)(31,59)(33,56)(34,62)(36,57)(37,39)(38,63)(42,61)(43,60)(46,58)(47,50)(52,64)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287261, #72287312, #72287334
maximal normal subgroup of	#72287261, #72287312, #72287334

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363785);