#125335

order	512
nilpotency class	4
isoclinism	isoclinic class of #125287
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,5)(4,6)(7,8)(9,10,13,17,14,19,30,37,15,18,12,25,27,36,28,39)(11,20,22,38,23,35,29,16,24,32,21,34,31,40,26,33);(2,5)(3,8)(6,7)(9,11,15,24)(10,16,18,33)(12,26,13,29)(14,31,27,23)(17,35,25,40)(19,38,36,34)(20,39,32,37)(21,30,22,28);(2,6)(5,7)(9,12,27,30,15,13,14,28)(10,17,36,39,18,25,19,37)(11,21,31,29,24,22,23,26)(16,20,38,40,33,32,34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32531);