#150514

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2328 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gh,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,d^c=dh,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, 57710);