#150813

order	512
nilpotency class	5
isoclinism	isoclinic class of #150812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11875
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 ◁ #2932 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,15,18,36)(4,21,23,49)(5,26,30,10)(7,14,34,46)(8,13,28,47)(9,22,37,56)(11,25,42,31)(16,51,27,38)(17,39,54,45)(19,43,57,62)(20,50,53,63)(24,59,33,44)(29,40,60,41)(32,58,35,61)(48,52,64,55);(1,3,8,20)(2,9,13,38)(4,22,24,16)(5,27,31,56)(6,32,28,57)(7,35,29,19)(10,36,42,63)(11,43,26,61)(12,45,47,64)(14,48,40,39)(15,49,50,44)(17,33,55,23)(18,34,53,60)(21,58,59,62)(25,52,30,54)(37,41,51,46);(1,4,7,25,8,24,29,30)(2,10,40,59,13,42,14,21)(3,16,19,52,20,22,35,54)(5,28,23,60,31,6,33,34)(9,36,48,58,38,63,39,62)(11,41,49,47,26,46,44,12)(15,45,61,37,50,64,43,51)(17,53,27,32,55,18,56,57)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58009);