#150618

order	512
nilpotency class	5
isoclinism	isoclinic class of #150615
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11808
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	#381	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

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

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, 57814);