#483033

order	512
nilpotency class	4
isoclinism	isoclinic class of #483032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#331
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60648	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1043 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1043 conjugacy classes of subgroups — too many to draw. Only its 312 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=h,f^2=gh,g^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390229);