#130649

order	512
nilpotency class	4
isoclinism	isoclinic class of #130474
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

922 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 922 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=gi,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,e^c=ei,f^c=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37845);