#503526

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33175	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

883 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 883 conjugacy classes of subgroups — too many to draw. Only its 178 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=ef,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410722);