#152138

order	512
nilpotency class	5
isoclinism	isoclinic class of #152136
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12707
commutator	#117	C₂²×C₄
Frattini, cosocle	#535	C₂³
prime radical	#3487	C₂²
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3487 ◁ G
LECS	G ▷ #535 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

651 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,13,17,40)(4,9)(5,22,23,56)(6,25,27,11)(7,12)(8,15,29,45)(10,24)(14,42,44,62)(16,36,26,41)(18,39,49,61)(19,50,51,55)(20,53,54,34)(21,35)(28,43,58,63)(30,46,31,47)(32,48,59,64)(33,52)(37,60)(38,57);(1,3,7,16)(2,8,12,31)(4,18,21,14)(5,15,23,47)(6,26,27,17)(9,32,35,28)(10,36,38,13)(11,29,25,30)(19,48,51,43)(20,44,54,49)(22,41,56,40)(24,45,57,46)(33,42,37,39)(34,59,53,58)(50,62,55,61)(52,64,60,63);(1,4,7,21)(2,9,12,35)(3,14,16,18)(5,19,23,51)(6,20,27,54)(8,28,31,32)(10,37,38,33)(11,34,25,53)(13,39,36,42)(15,43,47,48)(17,44,26,49)(22,55,56,50)(24,52,57,60)(29,58,30,59)(40,61,41,62)(45,63,46,64)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59334);