#150305

order	512
nilpotency class	5
isoclinism	isoclinic class of #150220
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	#391	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2311 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=i,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,d^c=di,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, 57501);