#149882

order	512
nilpotency class	5
isoclinism	isoclinic class of #149795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 256 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,6,5,7,3,8)(9,10,12,18,14,20,26,36)(11,21,23,38,22,16,33,34)(13,17,29,35,30,19,27,37)(15,25,31,28,32,39,40,24);(9,11)(10,15,20,32)(12,25)(13,28)(14,22)(16,17,21,19)(18,34,36,38)(23,29)(24,30)(26,39)(27,33)(31,37,40,35);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,23)(13,24)(14,22)(15,19)(17,32)(18,34)(20,21)(25,29)(26,33)(27,39)(28,30)(31,37)(35,40)(36,38)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dhi,e^b=efhi,f^b=fi,d^c=di,e^d=ehi,f^e=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, 57078);