#152250

order	512
nilpotency class	5
isoclinism	isoclinic class of #149791
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	#12798
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^c=eh,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59446);