#146700

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
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 ◁ C₂²×#131 ◁ 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)(4,21,24,42)(5,26,29,11)(6,31,30,12)(7,33)(8,14)(9,17,36,27)(10,23)(13,28)(16,46,49,64)(18,37,52,44)(19,35)(20,41)(22,39)(25,59,58,38)(32,40)(34,60,61,50)(43,48)(45,53)(47,62)(51,63)(54,56)(55,57);(1,3,7,19,8,20,28,53)(2,9,13,37,14,36,33,44)(4,16,25,50,24,49,58,60)(5,27,30,52,29,17,6,18)(10,34,39,46,40,61,55,64)(11,41,12,35,26,15,31,45)(21,54,38,63,42,62,59,43)(22,56,23,48,57,47,32,51);(1,4)(2,10)(3,16)(5,22)(6,32)(7,25)(8,24)(9,34)(11,38)(12,42)(13,39)(14,40)(15,43)(17,47)(18,51)(19,50)(20,49)(21,31)(23,30)(26,59)(27,56)(28,58)(29,57)(33,55)(35,62)(36,61)(37,46)(41,63)(44,64)(45,54)(48,52)(53,60)

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,g^a=gi,h^a=hi,d^b=di,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	#72303807, #72313027, #72313033
maximal normal subgroup of	#72303807, #72313027, #72313033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53896);