#128047

order	512
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 581 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) = 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)(17,18,21,26,23,28,22,27)(19,29,31,46,33,42,32,47)(20,25,35,43,36,44,37,45)(24,38,39,34,41,48,40,30);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,32,23,33,21,31)(18,24,27,40,28,41,26,39)(20,34,35,48,36,30,37,38)(25,42,43,47,44,29,45,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35243);