#128162

order	512
nilpotency class	4
isoclinism	isoclinic class of #128001
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	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ 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)(17,18,22,25,23,27,20,33)(19,28,31,44,32,42,29,45)(21,26,36,41,37,43,34,48)(24,38,39,35,40,46,47,30);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,24)(20,29)(21,35,37,30)(22,31)(23,32)(25,39)(26,42,43,28)(27,40)(33,47)(34,38,36,46)(41,45,48,44);(17,20,23,22)(18,25,27,33)(19,29,32,31)(21,34,37,36)(24,39,40,47)(26,41,43,48)(28,44,42,45)(30,46,35,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,e^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,d^b=dh,e^b=eg,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, 35358);