#150826

order	512
nilpotency class	5
isoclinism	isoclinic class of #150824
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11877
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 339 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ghi,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efg,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=eh,f^e=fi

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, 58022);