#493405

order	512
nilpotency class	4
isoclinism	isoclinic class of #493397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1781
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	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×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3239 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 3239 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,g^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400601);