#150163

order	512
nilpotency class	5
isoclinism	isoclinic class of #150163
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11678
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	#346	C₂³
prime radical	#2218	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

195 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225041, #72319310, #72319344, …
maximal normal subgroup of	#72225041, #72319310, #72319344, …
maximal quotient of	C₂×#150163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57359);