#150209

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

403 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,5)(4,7)(6,8)(9,10)(11,20,23,30)(12,16)(13,18,27,17)(14,19)(15,22,29,26)(21,34,36,40)(24,32,38,31)(25,33)(28,35,39,37);(2,5)(3,8)(6,7)(9,11,14,22)(10,15)(12,21,25,35)(13,26,27,23)(16,28)(17,20)(18,29)(19,30)(24,37,38,36)(31,34)(32,39)(33,40);(9,12,14,25)(10,16,19,33)(11,21,22,35)(13,24,27,38)(15,28,30,40)(17,31,18,32)(20,34,29,39)(23,36,26,37)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efg,f^b=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57405);