#128046

order	512
nilpotency class	4
isoclinism	isoclinic class of #128001
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

609 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=i,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, 35242);