#128839

order	512
nilpotency class	4
isoclinism	isoclinic class of #128836
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1762
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 444 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) = 64

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

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

Polycyclic presentation:

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

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, 36035);