#150550

order	512
nilpotency class	5
isoclinism	isoclinic class of #150429
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fhi,c^2=g,d^2=ghi,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cfhi,d^b=dg,e^b=efgh,d^c=dhi,f^c=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57746);