#128713

order	512
nilpotency class	4
isoclinism	isoclinic class of #128583
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8202
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₃₂ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

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=fg,g^a=gi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224946, #72224947, #72224960, …
maximal normal subgroup of	#72224946, #72224947, #72224960, …
maximal quotient of	C₂×#128713

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35909);