#138211

order	512
nilpotency class	4
isoclinism	isoclinic class of #138198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33175	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

503 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45407);