#141444

order	512
nilpotency class	4
isoclinism	isoclinic class of #141417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,8)(4,13)(5,15)(6,10)(7,11)(9,14)(12,16)(17,18,22,24,21,25,20,30)(19,26,28,31,29,32,27,23);(1,3,5,11,6,12,14,4)(2,7,9,8,10,13,15,16)(17,19)(18,23,25,31)(20,28,22,27)(21,29)(24,26)(30,32);(1,4,14,12,6,11,5,3)(2,8,15,7,10,16,9,13)(20,22)(24,30)(26,32)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,d^2=gh,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48640);