#137759

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
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	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 96 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)(17,18,22,29,24,27,31,46,23,28,41,47,21,37,39,43)(19,30,33,35,34,44,48,36,26,40,45,38,32,25,42,20);(1,3,5,10)(2,6)(4,9,11,15)(7,12)(8,13)(14,16)(17,19,31,48,21,32,22,33,23,26,39,42,24,34,41,45)(18,25,43,38,37,40,47,36,28,44,46,35,27,30,29,20);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20,23,36)(18,26,28,19)(21,38,24,35)(22,40,41,30)(25,39,44,31)(27,32,37,34)(29,33,47,45)(42,46,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44955);