#142520

order	512
nilpotency class	4
isoclinism	isoclinic class of #142472
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#3459	C₂²
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

637 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,e^c=eg,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49716);