#128695

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×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	#8200
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 311 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224949, #72224963, #72224964, …
maximal normal subgroup of	#72224949, #72224963, #72224964, …
maximal quotient of	C₂×#128695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35891);