#143000

order	512
nilpotency class	4
isoclinism	isoclinic class of #142972
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3493	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 437 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)(33,34)(35,44)(36,41)(37,40)(38,43)(39,45)(42,50)(46,57)(47,59)(48,54)(49,56)(51,64)(52,60)(53,61)(55,63)(58,62);(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)(33,35)(34,39)(36,45)(37,46)(38,47)(40,53)(41,44)(42,51)(43,52)(48,61)(49,60)(50,62)(54,57)(55,58)(56,59)(63,64);(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)(34,40)(37,50)(39,51)(41,54)(42,43)(44,57)(45,60)(46,62)(48,63)(52,53)(55,56)(58,59)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=i,f^2=g,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=fg,h^b=hi,d^c=dh,e^c=ei,f^c=fg,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, 50196);