#137757

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

361 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(17,18,22,27,23,28,42,46,24,29,43,47,21,26,31,32)(19,30,34,48,33,38,45,41,35,40,25,44,39,20,36,37);(1,3,5,10)(2,6)(4,9,11,15)(7,12)(8,13)(14,16)(17,19,31,45,21,39,43,34,24,35,42,36,23,33,22,25)(18,20,32,48,26,40,47,37,29,38,46,44,28,30,27,41);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20)(18,25)(19,32)(21,40)(22,41)(23,30)(24,38)(26,45)(27,33)(28,36)(29,34)(31,48)(35,46)(37,43)(39,47)(42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44953);