#474324

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16954
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

640 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)(5,11,28,39)(6,13)(7,12)(8,14)(9,18)(10,24)(16,34)(17,42,47,31)(19,32)(20,33)(22,30)(23,50,53,62)(25,37)(26,36)(27,40,29,38)(35,54,61,52)(41,45)(43,60,46,64)(44,56)(48,58,49,59)(51,55,63,57);(1,3)(2,9,14,15)(4,22)(5,17)(6,20)(7,19)(8,18)(10,34,37,56)(11,31,39,42)(12,33,13,32)(16,25)(21,41,36,30)(23,51)(24,44)(26,45)(27,49)(28,47)(29,48)(35,60,61,64)(38,59,40,58)(43,54)(46,52)(50,57,62,55)(53,63);(1,4,6,24,8,26,7,25)(2,10,12,21,14,37,13,36)(3,16,19,45,18,44,20,22)(5,23,27,52,28,53,29,54)(9,30,32,56,15,41,33,34)(11,35,38,50,39,61,40,62)(17,46,48,51,47,43,49,63)(31,57,58,60,42,55,59,64);(1,5,6,27,8,28,7,29)(2,11,12,38,14,39,13,40)(3,17,20,49,18,47,19,48)(4,23,24,52,26,53,25,54)(9,31,33,59,15,42,32,58)(10,35,21,50,37,61,36,62)(16,43,22,51,44,46,45,63)(30,55,34,60,41,57,56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381520);