#138296

order	512
nilpotency class	4
isoclinism	isoclinic class of #138296
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9149
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=gh,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,e^c=ei,f^d=fi

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, 45492);