#135328

order	512
nilpotency class	4
isoclinism	isoclinic class of #124660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1510
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8713
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,14,17,8)(4,19,20,51)(5,23,26,40)(6,12)(7,10)(9,21)(11,27,37,59)(13,29)(15,42)(16,46)(18,33,49,32)(22,52,28,39)(24,55)(25,34)(30,44,41,54)(31,48)(35,58)(36,53)(38,57)(43,60)(45,63)(47,62,56,61)(50,64);(1,3)(2,8)(4,15)(5,24)(6,18)(7,17)(9,30)(10,14)(11,38)(12,32)(13,33)(16,26)(19,42)(20,43)(21,54)(22,45)(23,46)(25,56)(27,57)(28,50)(29,49)(31,37)(34,61)(35,62)(36,41)(39,64)(40,55)(44,53)(47,58)(48,59)(51,60)(52,63);(1,4,6,28,7,20,29,22)(2,9,13,34,10,36,12,35)(3,15,18,50,17,43,49,45)(5,25,27,21,26,58,59,53)(8,30,33,61,14,41,32,62)(11,39,40,19,37,52,23,51)(16,47,48,44,24,56,57,54)(31,63,46,60,38,64,55,42)

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fgh,g^c=gi,h^c=hi,f^e=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, 42524);