#152996

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13252
prime radical	#33713	C₂
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

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

normal subgroups only

This group has 321 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225337, #72225343, #72225344, …
maximal normal subgroup of	#72225337, #72225343, #72225344, …
maximal quotient of	#72225478

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60192);