#502967

order	512
nilpotency class	4
isoclinism	isoclinic class of #501084
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, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

862 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,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, 410163);