#149744

order	512
nilpotency class	5
isoclinism	isoclinic class of #149743
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11499
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#381	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 983 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)(3,13,16,42,17,35,48,57)(4,9,19,32)(5,21,24,39)(7,12,26,40)(8,27,29,22,30,51,38,15)(10,34,37,25)(14,43,45,64,46,41,62,63)(18,36,49,60)(20,33,50,61)(23,31,52,58)(28,54,55,47,56,53,59,44);(1,3)(2,8)(4,14)(5,22)(6,16)(7,17)(9,28)(10,35)(11,38)(12,30)(13,21)(15,25)(18,47)(19,45)(20,46)(23,44)(24,51)(26,48)(27,34)(29,40)(31,41)(32,59)(33,56)(36,43)(37,42)(39,57)(49,53)(50,62)(52,54)(55,61)(58,64)(60,63);(1,4,7,20)(2,9,12,33)(3,14,17,46)(5,23,25,18)(6,19,26,50)(8,28,30,56)(10,36,21,31)(11,32,40,61)(13,41,35,43)(15,47,22,44)(16,45,48,62)(24,52,34,49)(27,53,51,54)(29,55,38,59)(37,60,39,58)(42,63,57,64)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=i,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=di,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	#72284174, #72284191, #72284224
maximal normal subgroup of	#72284174, #72284191, #72284224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56940);