#471050

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16741
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 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)(3,15,20,42)(4,21)(5,27,31,40)(6,12)(7,13)(8,14)(9,18,34,19)(10,24)(11,30,41,29)(16,32,47,53)(17,48)(22,43,46,35)(23,36,56,58)(25,38)(26,37)(28,39,57,54)(33,52)(44,51)(45,63)(49,60)(50,62)(55,64)(59,61);(1,3,7,18,8,20,6,19)(2,9,13,15,14,34,12,42)(4,22,24,47,26,46,25,16)(5,17,30,50,31,52,29,51)(10,35,37,32,38,43,21,53)(11,33,27,44,41,48,40,62)(23,55,28,59,56,49,57,45)(36,61,39,60,58,63,54,64);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,39)(12,21)(13,37)(14,38)(15,43)(17,49)(18,22)(19,46)(20,47)(23,29)(27,58)(30,56)(31,57)(33,61)(34,53)(35,42)(36,40)(41,54)(44,60)(45,50)(48,63)(51,59)(52,55)(62,64);(1,5,6,29,8,31,7,30)(2,11,13,27,14,41,12,40)(3,17,19,51,20,52,18,50)(4,23,24,28,26,56,25,57)(9,33,15,44,34,48,42,62)(10,36,21,54,38,58,37,39)(16,45,22,55,47,59,46,49)(32,60,35,61,53,64,43,63)

Polycyclic presentation:

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

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, 378246);