#152247

order	512
nilpotency class	5
isoclinism	isoclinic class of #149792
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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

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

normal subgroups only

This group has 297 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)(3,14)(4,19,22,34)(5,23)(6,12,27,11)(7,13)(8,16)(9,21,33,26)(10,25)(15,41,44,62)(17,39,48,38)(18,40)(20,51,53,32)(24,35,56,36)(28,57,58,64)(29,45,30,60)(31,46)(37,43,61,47)(42,49,63,59)(50,55,54,52);(1,3,5,16,7,18,25,46)(2,8,10,14,13,31,23,40)(4,15,20,42,21,43,52,57)(6,17,24,45,27,48,56,60)(9,28,32,37,34,59,54,62)(11,29,35,38,12,30,36,39)(19,49,50,41,33,58,51,61)(22,44,53,63,26,47,55,64);(1,4,7,21)(2,9)(3,15,18,43)(5,20,25,52)(6,26,27,22)(8,28)(10,32)(11,19)(12,33)(13,34)(14,37)(16,42,46,57)(17,47,48,44)(23,54)(24,55,56,53)(29,49)(30,58)(31,59)(35,50)(36,51)(38,41)(39,61)(40,62)(45,64,60,63)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,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, 59443);