#129013

order	512
nilpotency class	4
isoclinism	isoclinic class of #128808
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1762
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#466	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

266 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,d^c=di,f^c=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, 36209);