#146673

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1001 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 188 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)(3,15)(4,21,24,40)(5,26,28,11)(6,30,29,12)(7,13)(8,14)(9,17,34,51)(10,23)(16,45,48,62)(18,35,52,42)(19,43)(20,44)(22,56)(25,55,58,36)(27,39)(31,38)(32,49,59,64)(33,50)(37,57)(41,47)(46,60)(53,61)(54,63);(1,3,7,19,8,20,27,50)(2,9,13,35,14,34,39,42)(4,16,25,49,24,48,58,64)(5,17,29,52,28,51,6,18)(10,32,37,45,38,59,56,62)(11,15,30,43,26,44,12,33)(21,54,55,61,40,60,36,41)(22,46,23,47,57,63,31,53);(1,4)(2,10)(3,16)(5,22)(6,31)(7,25)(8,24)(9,32)(11,36)(12,40)(13,37)(14,38)(15,41)(17,46)(18,53)(19,49)(20,48)(21,30)(23,29)(26,55)(27,58)(28,57)(33,60)(34,59)(35,45)(39,56)(42,62)(43,54)(44,61)(47,52)(50,64)(51,63)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317336, #72317344, #72317364
maximal normal subgroup of	#72317336, #72317344, #72317364

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53869);