#128968

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂²	C₄×#99
prime radical	C₂³	C₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 347 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36164);