#473992

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16892
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

746 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)(3,15,20,35)(4,21)(5,26,29,41)(6,12)(7,13)(8,14)(9,18,34,19)(10,24)(11,30,42,28)(16,36,22,32)(17,48)(23,55,56,37)(25,38)(27,59,57,40)(31,39)(33,52)(43,46,53,47)(44,50)(45,63)(49,60)(51,58)(54,61)(62,64);(1,3,7,18,8,20,6,19)(2,9,13,15,14,34,12,35)(4,22,24,46,25,16,31,47)(5,17,28,51,29,52,30,50)(10,36,38,43,39,32,21,53)(11,33,41,44,42,48,26,58)(23,54,57,64,56,45,27,49)(37,62,59,63,55,60,40,61);(1,4,6,24,8,25,7,31)(2,10,13,21,14,39,12,38)(3,16,19,47,20,22,18,46)(5,27,28,56,29,57,30,23)(9,32,15,43,34,36,35,53)(11,40,26,37,42,59,41,55)(17,49,51,45,52,64,50,54)(33,61,58,62,48,63,44,60);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,33)(10,37)(12,26)(13,41)(14,42)(15,44)(16,45)(18,51)(19,50)(20,52)(21,40)(22,54)(24,57)(25,56)(27,31)(32,60)(34,48)(35,58)(36,62)(38,59)(39,55)(43,63)(46,64)(47,49)(53,61)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,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, 381188);