#150149

order	512
nilpotency class	5
isoclinism	isoclinic class of #150148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11667
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	#346	C₂³
prime radical	C₂²×D₁₆	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

235 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=hi,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=ch,d^b=dfh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225042, C₂×#150149, #72343285, …
maximal normal subgroup of	#72225042, C₂×#150149, #72343285, …
maximal quotient of	C₂×#150149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57345);