#510571

order	512
nilpotency class	4
isoclinism	isoclinic class of #510571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31736
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=fh,c^2=eh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417767);