#128849

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

421 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36045);