#493399

order	512
nilpotency class	4
isoclinism	isoclinic class of #493397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1781
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62848	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3259 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 3259 conjugacy classes of subgroups — too many to draw. Only its 328 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)(3,13,17,37)(4,9)(5,10)(6,24,27,11)(7,12)(8,16)(14,41,44,39)(15,40,47,60)(18,31)(19,38,26,32)(20,33)(21,34)(22,51,53,35)(23,36)(25,30)(28,43)(29,46)(42,62,45,61)(48,55)(49,59,50,58)(52,57)(54,63)(56,64);(1,3)(2,8)(4,14)(5,15)(6,25)(7,17)(9,28)(10,29)(11,37)(12,30)(13,24)(16,27)(18,42)(19,48)(20,44)(21,45)(22,52)(23,47)(26,43)(31,54)(32,39)(33,55)(34,56)(35,60)(36,57)(38,41)(40,51)(46,53)(49,64)(50,63)(58,61)(59,62);(1,4,7,20)(2,9,12,33)(3,14,17,44)(5,21,23,18)(6,26,27,19)(8,28,30,55)(10,34,36,31)(11,38,24,32)(13,39,37,41)(15,45,47,42)(16,48,25,43)(22,49,53,50)(29,56,57,54)(35,58,51,59)(40,62,60,61)(46,63,52,64);(1,5,7,23)(2,10,12,36)(3,15,17,47)(4,18,20,21)(6,22,27,53)(8,29,30,57)(9,31,33,34)(11,35,24,51)(13,40,37,60)(14,42,44,45)(16,46,25,52)(19,49,26,50)(28,54,55,56)(32,58,38,59)(39,61,41,62)(43,63,48,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303039, #72303150, #72303154, …
maximal normal subgroup of	#72303039, #72303150, #72303154, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400595);