#150842

order	512
nilpotency class	5
isoclinism	isoclinic class of #150842
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11878
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

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

normal subgroups only

This group has 255 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)(3,15,18,39)(4,21,23,46)(5,26,25,42)(7,14,32,45)(8,13,31,48)(9,16,36,27)(10,29,11,43)(17,49,54,47)(19,44,30,63)(20,51,52,35)(22,59,56,37)(24,58,28,50)(33,61,62,57)(34,40,60,41)(38,53,64,55);(1,3)(2,9,13,37)(4,22)(5,27)(6,30)(7,33)(8,20)(10,39,42,35)(11,44,26,61)(12,47,48,64)(14,49,40,38)(15,50,51,46)(16,24)(17,28)(18,32)(19,34)(21,57,58,63)(23,55)(25,53)(29,56)(31,62)(36,45,59,41)(43,54)(52,60);(1,4,7,25,8,24,34,43)(2,10,40,58,13,42,14,21)(3,16,19,53,20,22,33,54)(5,6,28,32,29,31,23,60)(9,35,49,63,37,39,38,57)(11,45,50,12,26,41,46,48)(15,47,61,59,51,64,44,36)(17,52,27,62,55,18,56,30)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=g,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=efgi,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, 58038);