#150620

order	512
nilpotency class	5
isoclinism	isoclinic class of #150586
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 417 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=dg,e^b=efghi,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, 57816);