#491348

order	512
nilpotency class	4
isoclinism	isoclinic class of #491245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

379 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,7,12,21)(4,14,16,24)(5,17,19,9)(8,18,23,15)(11,25,26,32)(13,29,28,22)(20,30,31,27)(33,34,38,43)(35,44,48,61)(36,41,39,50)(37,51,52,55)(40,54,57,47)(42,45,58,53)(46,59,49,64)(56,63,60,62);(1,3)(2,7)(4,11)(5,13)(6,12)(8,20)(9,22)(10,21)(14,25)(15,27)(16,26)(17,29)(18,30)(19,28)(23,31)(24,32)(33,35,45,62,39,49,37,47,38,48,53,63,36,46,52,54)(34,40,55,64,41,56,42,44,43,57,51,59,50,60,58,61);(1,4,6,16)(2,8,10,23)(3,11,12,26)(5,18,19,15)(7,20,21,31)(9,24,17,14)(13,30,28,27)(22,32,29,25)(33,34)(35,40)(36,50)(37,42)(38,43)(39,41)(44,47)(45,55)(46,60)(48,57)(49,56)(51,53)(52,58)(54,61)(59,63)(62,64);(3,12)(7,21)(11,26)(13,28)(20,31)(22,29)(25,32)(27,30)(33,36,38,39)(34,41,43,50)(35,46,48,49)(37,45,52,53)(40,56,57,60)(42,51,58,55)(44,59,61,64)(47,62,54,63)

Polycyclic presentation:

2^9|a^2=hi,b^2=egi,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=dh,d^c=di,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, 398544);