#125410

order	512
nilpotency class	4
isoclinism	isoclinic class of #125409
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

250 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32606);