#146583

order	512
nilpotency class	4
isoclinism	isoclinic class of #146581
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10735
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1585 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1585 conjugacy classes of subgroups — too many to draw. Only its 200 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)(3,13,16,33)(4,9)(5,10,22,35)(6,24,26,11)(7,12)(8,15)(14,40,42,38)(17,39)(18,30,21,34)(19,50,52,31)(20,32)(23,37)(25,36)(27,41)(28,45)(29,48,53,46)(43,60)(44,54)(47,56)(49,58)(51,57)(55,63,64,61)(59,62);(1,3,6,17,7,16,26,47)(2,8,11,29,12,28,24,53)(4,14,19,43,20,42,52,62)(5,15,23,46,22,45,25,48)(9,27,31,55,32,54,50,64)(10,33,36,56,35,13,37,39)(18,41,49,61,21,44,51,63)(30,38,57,59,34,40,58,60);(1,4,7,20)(2,9,12,32)(3,14,16,42)(5,21,22,18)(6,19,26,52)(8,27,28,54)(10,34,35,30)(11,31,24,50)(13,38,33,40)(15,44,45,41)(17,43,47,62)(23,51,25,49)(29,55,53,64)(36,58,37,57)(39,59,56,60)(46,63,48,61)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303181, #72303186, #72303196, …
maximal normal subgroup of	#72303181, #72303186, #72303196, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53779);