#503336

order	512
nilpotency class	4
isoclinism	isoclinic class of #501156
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, prime radical, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

866 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi

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, 410532);