#152374

order	512
nilpotency class	5
isoclinism	isoclinic class of #13717
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2233
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12867
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=dh,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^b=ei,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59570);