#502664

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	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

996 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,7)(4,6)(5,8)(9,10,14,12)(11,15,19,18)(13,20,21,16)(17,22,24,23);(1,3,4,8)(2,5,6,7)(11,17)(13,21)(15,22)(16,20)(18,23)(19,24);(9,11)(10,15)(12,18)(13,17,21,24)(14,19)(16,22,20,23);(2,6)(5,7)(9,12,14,10)(11,18,19,15)(13,20,21,16)(17,23,24,22)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,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, 409860);