#142503

order	512
nilpotency class	4
isoclinism	isoclinic class of #142484
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	#537	D₄
LCS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 464 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,c^b=cf,d^b=dg,f^b=fi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,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, 49699);