#135910

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8845
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

290 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fi,g^b=gi,d^c=dg,e^c=egi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225419, #72225466, #72225467
maximal normal subgroup of	#72225419, #72225466, #72225467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43106);