#140214

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9560
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,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=egi,g^a=gi,c^b=cgh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320710, #72320727, #72320773
maximal normal subgroup of	#72320710, #72320727, #72320773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47410);