#146202

order	512
nilpotency class	4
isoclinism	isoclinic class of #146200
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dei,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53398);