#491328

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	#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

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

normal subgroups only

This group has 398 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)(33,34,39,43)(35,44,46,58)(36,41,38,53)(37,51,52,64)(40,54,55,48)(42,50,57,45)(47,59,49,63)(56,61,60,62);(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)(33,35,45,61,38,49,52,54,39,46,50,62,36,47,37,48)(34,40,51,59,41,56,57,58,43,55,64,63,53,60,42,44);(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)(40,55)(44,58)(47,49)(48,54)(56,60)(59,63)(61,62);(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)(40,56,55,60)(42,51,57,64)(44,59,58,63)(48,62,54,61)

Polycyclic presentation:

2^9|a^2=hi,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, 398524);