#150829

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	#28495	C₂
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

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

normal subgroups only

This group has 342 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=ghi,d^2=gi,e^2=i,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, 58025);