#141543

order	512
nilpotency class	4
isoclinism	isoclinic class of #141451
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 378 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=eghi,d^2=ghi,e^2=ghi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48739);