#143004

order	512
nilpotency class	4
isoclinism	isoclinic class of #142975
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#534	D₄
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

425 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,h^b=hi,d^c=dh,e^c=ei,f^c=fgi,f^d=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50200);