#149806

order	512
nilpotency class	5
isoclinism	isoclinic class of #149783
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×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#384	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2285 ◁ G
LECS	G ▷ #384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

302 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=efh,e^d=ehi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57002);