#512100

order	512
nilpotency class	4
isoclinism	isoclinic class of #511751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33520	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1041 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 164 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,f^c=fh,g^c=ghi,h^c=hi,f^d=fhi,g^d=gh,h^d=hi

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, 419296);