#152874

order	512
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2226
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13177
prime radical	C₂³	#510
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 183 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225337, #72225342, #72225343, …
maximal normal subgroup of	#72225337, #72225342, #72225343, …
maximal quotient of	#72225493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60070);