#491327

order	512
nilpotency class	4
isoclinism	isoclinic class of #491227
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 ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 387 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)(38,53)(39,43)(40,48)(42,45)(46,57)(47,58)(49,63)(50,56)(52,64)(54,61)(55,62)(59,60);(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)(33,35,45,60,38,49,52,61,39,46,50,62,36,47,37,48)(34,40,51,58,41,55,56,57,43,54,64,63,53,59,42,44);(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)(35,46)(40,54)(44,57)(47,49)(48,61)(55,59)(58,63)(60,62);(3,12)(7,21)(11,26)(13,28)(20,31)(22,29)(25,32)(27,30)(33,36,39,38)(34,41,43,53)(35,47,46,49)(37,50,52,45)(40,55,54,59)(42,51,56,64)(44,58,57,63)(48,62,61,60)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,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,c^b=ci,d^b=dh,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, 398523);