#152956

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ei,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, 60152);