#483306

order	512
nilpotency class	4
isoclinism	isoclinic class of #483305
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1555
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60648	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

965 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 965 conjugacy classes of subgroups — too many to draw. Only its 208 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390502);