#139729

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

707 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,7,8,5)(4,6)(9,10,15,19,16,20,14,31)(11,13,18,29,24,30,23,21)(12,25,27,17,28,34,32,35)(22,36,33,26,37,39,38,40);(1,2)(3,5)(4,6)(7,8)(9,11,21,10)(12,22,36,17)(13,19,14,23)(15,18,30,31)(16,24,29,20)(25,27,33,40)(26,34,32,38)(28,37,39,35);(1,3)(2,5)(4,8)(6,7)(9,12,16,28)(10,17,20,35)(11,22,24,37)(13,26,30,40)(14,32,15,27)(18,33,23,38)(19,34,31,25)(21,36,29,39)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46925);