#148028

order	512
nilpotency class	4
isoclinism	isoclinic class of #148027
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11043
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

773 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 773 conjugacy classes of subgroups — too many to draw. Only its 132 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)(3,13)(4,18,22,37)(5,23)(6,27,29,11)(7,12)(8,15)(9,21)(10,26)(14,41,19,52)(16,47,49,39)(17,40)(20,53,24,51)(25,36,57,55)(28,34)(30,59)(31,45,58,64)(32,46)(33,56)(35,54)(38,43)(42,62,44,50)(48,60)(61,63);(1,3,5,15,7,17,26,46)(2,8,10,13,12,32,23,40)(4,19,20,44,22,14,24,42)(6,16,25,45,29,49,57,64)(9,30,33,60,34,61,35,38)(11,31,36,39,27,58,55,47)(18,50,51,52,37,62,53,41)(21,48,54,59,28,43,56,63);(1,4,7,22)(2,9,12,34)(3,14,17,19)(5,24,26,20)(6,28,29,21)(8,30,32,61)(10,35,23,33)(11,37,27,18)(13,38,40,60)(15,44,46,42)(16,48,49,43)(25,54,57,56)(31,62,58,50)(36,51,55,53)(39,52,47,41)(45,63,64,59)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=ghi,g^e=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55224);