#152955

order	512
nilpotency class	5
isoclinism	isoclinic class of #152953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13224
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60151);