#432845

order	512
nilpotency class	3
isoclinism	isoclinic class of #432844
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15467
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1509 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1509 conjugacy classes of subgroups — too many to draw. Only its 188 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)(3,8,15,31)(4,9)(5,20)(6,24)(7,11)(10,23)(12,28)(13,33,42,29)(14,44,21,53)(16,38,25,32)(17,48)(18,39)(19,35)(22,40)(26,36)(27,37)(30,47,57,45)(34,51)(41,61,49,64)(43,56,46,58)(50,54)(52,60)(55,62,59,63);(1,3)(2,8,11,31)(4,13)(5,21,22,14)(6,25)(7,15)(9,33,35,29)(10,30)(12,38,24,32)(16,28)(17,49,50,41)(18,46)(19,42)(20,53)(23,45,27,47)(26,43)(34,59)(36,56,39,58)(37,57)(40,44)(48,64)(51,62,52,63)(54,61)(55,60);(1,4,7,19)(2,9,11,35)(3,13,15,42)(5,17,22,50)(6,26,28,18)(8,29,31,33)(10,34,37,60)(12,39,24,36)(14,41,21,49)(16,46,25,43)(20,54,40,48)(23,52,27,51)(30,55,57,59)(32,58,38,56)(44,61,53,64)(45,63,47,62);(1,5)(2,10)(3,14)(4,17)(6,27)(7,22)(8,30)(9,34)(11,37)(12,40)(13,41)(15,21)(16,47)(18,52)(19,50)(20,24)(23,28)(25,45)(26,51)(29,55)(31,57)(32,53)(33,59)(35,60)(36,54)(38,44)(39,48)(42,49)(43,63)(46,62)(56,61)(58,64)

Polycyclic presentation:

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

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, 340041);