#129170

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#442
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8285
prime radical	C₂³	#444
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

229 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,13,16,41,18,42,17,36)(4,19,22,53)(5,24)(6,12)(7,11)(8,15,30,48,31,49,32,25)(9,23,35,59)(10,27)(14,43,45,61,46,64,50,40)(20,39)(21,56,26,52)(28,38)(29,60,62,44,63,57,51,47)(33,55)(34,54,37,58);(1,3)(2,8,11,31)(4,14)(5,25)(6,16)(7,18)(9,29,35,63)(10,36,24,41)(12,30,38,32)(13,39,42,33)(15,20)(17,28)(19,51,53,62)(21,57)(22,46)(23,50)(26,60)(27,48)(34,64,37,43)(40,52,61,56)(44,58)(45,59)(47,54)(49,55);(1,4,20,54,28,59,5,26,7,22,55,58,6,23,27,21)(2,9,33,56,38,53,10,37,11,35,39,52,12,19,24,34)(3,14,15,47,17,45,25,60,18,46,49,44,16,50,48,57)(8,29,13,40,32,62,36,43,31,63,42,61,30,51,41,64)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224955, #72224964, #72224965, …
maximal normal subgroup of	#72224955, #72224964, #72224965, …
maximal quotient of	C₂×#129170

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36366);