#135261

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1823
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8706
prime radical	C₂⁴	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42457);