#128463

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#99
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	C₈×#99
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=di,f^a=fgi,d^b=dg,g^b=gi,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319273, #72343176, #72343179
maximal normal subgroup of	#72319273, #72343176, #72343179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35659);