#129102

order	512
nilpotency class	3
isoclinism	isoclinic class of #108534
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₄⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1767
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36298);