#450636

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15973
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

808 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)(3,14)(4,9,24,38)(5,26,30,40)(6,12)(7,13)(8,17)(10,28,41,29)(11,21)(15,43,46,35)(16,48,49,32)(18,33)(19,34)(20,31,47,59)(22,52)(23,42,25,37)(27,39,50,44)(36,53)(45,58)(51,60)(54,62)(55,56)(57,63)(61,64);(1,3,7,18)(2,8)(4,20,24,47)(5,27)(6,17,21,19)(9,35)(10,39,41,44)(11,14)(12,33)(13,34)(15,23,46,25)(16,29)(22,45)(26,32,40,48)(28,49)(30,50)(31,37)(36,61,62,58)(38,43)(42,59)(51,54)(52,63,56,60)(53,57)(55,64);(1,4,21,25,7,24,6,23)(2,9,11,37,13,38,12,42)(3,15,19,47,18,46,17,20)(5,22,28,53,30,55,29,54)(8,31,14,43,34,59,33,35)(10,36,26,56,41,62,40,52)(16,45,27,57,49,64,50,51)(32,58,44,63,48,61,39,60);(1,5)(2,10)(3,16)(4,22)(6,29)(7,30)(8,32)(9,36)(11,26)(12,40)(13,41)(14,44)(15,45)(17,50)(18,49)(19,27)(20,51)(21,28)(23,54)(24,55)(25,53)(31,58)(33,39)(34,48)(35,60)(37,56)(38,62)(42,52)(43,63)(46,64)(47,57)(59,61)

Polycyclic presentation:

2^9|a^2=efh,b^2=f,c^2=ef,e^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,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357832);