#466413

order	512
nilpotency class	3
isoclinism	isoclinic class of #466412
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

869 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)(5,27,31,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,29,42,30)(16,32)(17,33,50,60)(19,35)(20,34)(22,36)(23,51,53,37)(24,39)(26,38)(28,40,52,62)(43,49,59,48)(44,61,54,58)(45,56)(46,57)(47,64,63,55);(1,3)(2,9,14,35)(4,22,26,45)(5,17)(6,18)(7,19)(8,20)(10,36)(11,33,42,60)(12,15,13,34)(16,25,46,24)(21,32)(23,47,53,63)(27,43,41,59)(28,54,52,44)(29,48)(30,49)(31,50)(37,61)(38,57)(39,56)(40,55)(51,58)(62,64);(1,4,7,25,8,26,6,24)(2,10,13,21,14,39,12,38)(3,16,19,22,20,46,18,45)(5,28,30,23,31,52,29,53)(9,32,34,56,35,57,15,36)(11,40,27,37,42,62,41,51)(17,47,49,44,50,63,48,54)(33,58,43,55,60,61,59,64);(1,5,7,29,8,31,6,30)(2,11,13,41,14,42,12,27)(3,17,19,48,20,50,18,49)(4,23,25,28,26,53,24,52)(9,33,34,59,35,60,15,43)(10,37,21,40,39,51,38,62)(16,44,22,47,46,54,45,63)(32,55,56,58,57,64,36,61)

Polycyclic presentation:

2^9|a^2=ef,b^2=fg,c^2=ei,d^2=f,e^2=i,f^2=i,g^2=i,h^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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319592, #72320000, #72320069
maximal normal subgroup of	#72319592, #72320000, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373609);