#466063

order	512
nilpotency class	3
isoclinism	isoclinic class of #466062
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1009 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)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,31)(16,32)(17,43)(19,35)(20,34)(22,36)(23,40)(24,39)(26,38)(28,37)(29,41)(30,42)(33,49)(44,63)(45,57)(46,58)(47,59)(48,53)(50,60)(51,54)(52,62)(55,56)(61,64);(1,3)(2,9,14,35)(4,22,26,45)(5,17)(6,18)(7,19)(8,20)(10,36)(11,33,27,53)(12,15,13,34)(16,25,46,24)(21,32)(23,47,51,64)(28,55,52,44)(29,48)(30,49)(31,50)(37,61)(38,58)(39,57)(40,56)(41,60,42,43)(54,63)(59,62);(1,4,7,25,8,26,6,24)(2,10,13,21,14,39,12,38)(3,16,19,22,20,46,18,45)(5,28,30,23,31,52,29,51)(9,32,34,57,35,58,15,36)(11,40,42,37,27,54,41,62)(17,47,49,44,50,64,48,55)(33,59,43,56,53,61,60,63);(1,5,7,29,8,31,6,30)(2,11,13,41,14,27,12,42)(3,17,19,48,20,50,18,49)(4,23,25,28,26,51,24,52)(9,33,34,60,35,53,15,43)(10,37,21,40,39,62,38,54)(16,44,22,47,46,55,45,64)(32,56,57,59,58,63,36,61)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287261, #72287302, #72287345
maximal normal subgroup of	#72287261, #72287302, #72287345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373259);