#128820

order	512
nilpotency class	4
isoclinism	isoclinic class of #11796
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

380 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,7)(4,6)(5,8)(9,10,14,18,15,19,26,12)(11,20,23,36,24,37,38,39)(13,28,29,27,30,17,33,34)(16,31,21,35,32,40,25,22);(2,6)(3,4)(7,8)(9,11)(10,16)(12,25)(13,22,30,35)(14,23)(15,24)(17,20,28,37)(18,21)(19,32)(26,38)(27,39,34,36)(29,31,33,40);(1,3,5,4)(2,7,8,6)(9,12,26,19,15,18,14,10)(11,21,38,16,24,25,23,32)(13,27,33,28,30,34,29,17)(20,35,39,31,37,22,36,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh,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, 36016);