#152248

order	512
nilpotency class	5
isoclinism	isoclinic class of #149791
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)(4,19,23,36)(5,24)(6,12,28,11)(7,13)(8,16)(9,22,35,27)(10,26)(15,43,46,62)(17,41,50,40)(18,42)(20,53,49,39)(21,55,56,34)(25,37,59,38)(29,54,51,64)(30,47,31,61)(32,48)(33,45,60,63)(44,52,58,57);(1,3,5,16,7,18,26,48)(2,8,10,14,13,32,24,42)(4,20,21,54,22,15,44,45)(6,17,25,47,28,50,59,61)(9,33,34,62,36,29,57,39)(11,30,37,40,12,31,38,41)(19,51,52,53,35,60,55,43)(23,49,56,64,27,46,58,63);(1,4,7,22)(2,9)(3,15,18,20)(5,21,26,44)(6,27,28,23)(8,29)(10,34)(11,19)(12,35)(13,36)(14,39)(16,45,48,54)(17,49,50,46)(24,57)(25,58,59,56)(30,60)(31,51)(32,33)(37,52)(38,55)(40,43)(41,53)(42,62)(47,64,61,63)

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,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, 59444);