#130735

order	512
nilpotency class	4
isoclinism	isoclinic class of #130734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#61030	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

847 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 847 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,f^c=fg

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, 37931);