#491344

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	#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

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

normal subgroups only

This group has 390 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)(35,44)(36,41)(37,51)(38,50)(39,43)(40,47)(42,52)(45,57)(46,58)(48,64)(49,60)(53,54)(55,61)(56,63)(59,62);(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)(33,35,45,61,38,48,37,47,39,49,52,62,36,46,53,63)(34,40,54,64,41,55,42,44,43,56,51,58,50,59,57,60);(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)(38,41)(39,43)(44,47)(45,54)(46,59)(48,55)(49,56)(51,52)(53,57)(58,62)(60,63)(61,64);(3,12)(7,21)(11,26)(13,28)(20,31)(22,29)(25,32)(27,30)(33,36,39,38)(34,41,43,50)(35,46,49,48)(37,45,53,52)(40,55,56,59)(42,51,57,54)(44,58,60,64)(47,61,63,62)

Polycyclic presentation:

2^9|a^2=h,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, 398540);