#140775

order	512
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#493
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9663
prime radical	#3414	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gi,e^2=gi,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=ch,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224956, #72224960, #72224964, …
maximal normal subgroup of	#72224956, #72224960, #72224964, …
maximal quotient of	C₂×#140775

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47971);