#135422

order	512
nilpotency class	4
isoclinism	isoclinic class of #124690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1548
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8724
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=hi,f^2=gh,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fgh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42618);