#146200

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

Polycyclic presentation:

2^9|a^2=d,b^2=de,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, 53396);