#512614

order	512
nilpotency class	5
isoclinism	isoclinic class of #512613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
prime radical	#60648	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1069 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1069 conjugacy classes of subgroups — too many to draw. Only its 192 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)(3,14,17,8)(4,9)(5,10)(6,12,26,11)(7,13)(15,39,42,27)(16,40,45,28)(18,29,25,30)(19,31)(20,33,49,32)(21,34)(22,35)(23,37,52,36)(24,38)(41,61,63,53)(43,54,48,55)(44,62,64,56)(46,57,51,58)(47,60,50,59);(1,3)(2,8)(4,15)(5,16)(6,25)(7,18)(9,27)(10,28)(11,29)(12,14)(13,30)(17,26)(19,41)(20,48)(21,43)(22,44)(23,51)(24,46)(31,53)(32,54)(33,39)(34,55)(35,56)(36,57)(37,40)(38,58)(42,49)(45,52)(47,64)(50,63)(59,62)(60,61);(1,4,7,21)(2,9,13,34)(3,15,18,43)(5,22,24,19)(6,20,26,49)(8,27,30,55)(10,35,38,31)(11,32,12,33)(14,39,29,54)(16,44,46,41)(17,42,25,48)(23,50,52,47)(28,56,58,53)(36,60,37,59)(40,62,57,61)(45,64,51,63);(1,5,7,24)(2,10,13,38)(3,16,18,46)(4,19,21,22)(6,23,26,52)(8,28,30,58)(9,31,34,35)(11,36,12,37)(14,40,29,57)(15,41,43,44)(17,45,25,51)(20,47,49,50)(27,53,55,56)(32,59,33,60)(39,61,54,62)(42,63,48,64)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,f^e=fhi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419810);