#152886

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#127
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13190
prime radical	#525	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 288 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,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,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217181, #72224997, #72225005, …
maximal normal subgroup of	#72217181, #72224997, #72225005, …
maximal quotient of	C₂×#152886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60082);