#139791

order	512
nilpotency class	4
isoclinism	isoclinic class of #139712
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

1145 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 1145 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,7,8,5)(4,6)(9,10)(11,13,18,23)(12,19,21,15)(14,16)(17,22)(20,24);(1,2)(3,5)(4,6)(7,8)(9,11,13,16,14,18,23,10)(12,17,22,19,21,24,20,15);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,17)(13,22)(14,21)(16,19)(18,24)(20,23)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46987);