#153003

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#391
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13264
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

207 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,d^2=i,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217181, #72224999, #72225007, …
maximal normal subgroup of	#72217181, #72224999, #72225007, …
maximal quotient of	C₂×#153003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60199);