#139570

order	512
nilpotency class	4
isoclinism	isoclinic class of #139568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#24279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,7,8,5)(4,6)(9,10,14,19,16,12,15,27)(11,20,24,36,25,37,23,35)(13,28,29,26,31,40,30,18)(17,32,34,39,21,38,33,22);(1,2)(3,5)(4,6)(7,8)(9,11,13,22,14,24,29,32,16,25,31,39,15,23,30,38)(10,17,26,37,27,33,28,36,12,21,18,20,19,34,40,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,21)(13,18)(14,19)(15,27)(17,25)(20,22)(23,33)(24,34)(26,31)(28,30)(29,40)(32,35)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,e^b=eh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46766);