#128170

order	512
nilpotency class	4
isoclinism	isoclinic class of #128009
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,22,25,23,27,20,32)(19,28,29,44,31,45,36,43)(21,26,34,41,35,42,33,48)(24,37,39,30,40,46,38,47);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,24)(20,29)(21,30,35,47)(22,36)(23,31)(25,38)(26,28,42,45)(27,40)(32,39)(33,46,34,37)(41,43,48,44);(17,20,23,22)(18,25,27,32)(19,29,31,36)(21,33,35,34)(24,38,40,39)(26,41,42,48)(28,43,45,44)(30,46,47,37)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,d^b=dh,e^b=egi,h^b=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35366);