#149754

order	512
nilpotency class	5
isoclinism	isoclinic class of #149753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11500
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#381	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

391 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=i,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,e^b=efi,g^b=gi,d^c=di,e^d=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284166, #72284199, #72284224
maximal normal subgroup of	#72284166, #72284199, #72284224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56950);