#512075

order	512
nilpotency class	4
isoclinism	isoclinic class of #512074
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32979
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

931 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 931 conjugacy classes of subgroups — too many to draw. Only its 172 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,h^c=hi,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419271);