#490396

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
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	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

645 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397592);