#490558

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1677
prime radical	#33594	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=eh,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,f^b=fi,e^c=ei,f^c=fh,h^c=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, 397754);