#121300

order	512
nilpotency class	3
isoclinism	isoclinic class of #121298
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

355 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fi,g^b=gi,e^c=ehi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28496);