#151682

order	512
nilpotency class	5
isoclinism	isoclinic class of #151682
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12468
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#12436	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2170 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

127 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 127 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=dfgh,d^2=fh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eg,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58878);