#121375

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

422 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28571);