#488676

order	512
nilpotency class	4
isoclinism	isoclinic class of #488553
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

637 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,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, 395872);