#490372

order	512
nilpotency class	4
isoclinism	isoclinic class of #490075
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 ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

653 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 653 conjugacy classes of subgroups — too many to draw. Only its 142 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)(33,34,39,43)(35,44,47,62)(36,41,38,53)(37,49,52,63)(40,55,58,46)(42,48,60,51)(45,61,54,59)(50,56,64,57);(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)(33,35)(34,40)(36,45)(37,50)(38,54)(39,47)(41,56)(42,59)(43,58)(44,49)(46,51)(48,55)(52,64)(53,57)(60,61)(62,63);(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)(33,36,39,38)(34,41,43,53)(35,45,47,54)(37,48,52,51)(40,56,58,57)(42,49,60,63)(44,61,62,59)(46,50,55,64);(3,15)(8,23)(13,24)(14,29)(16,17)(22,28)(27,32)(30,31)(33,37,36,48,39,52,38,51)(34,42,53,63,43,60,41,49)(35,46,54,64,47,55,45,50)(40,44,56,61,58,62,57,59)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397568);