#140752

order	512
nilpotency class	4
isoclinism	isoclinic class of #13247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1972
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9659
prime radical	C₂×#508	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

309 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,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,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224946, #72224947, #72224949, …
maximal normal subgroup of	#72224946, #72224947, #72224949, …
maximal quotient of	#72227212, C₂×#140752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47948);