#150213

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	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 ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 395 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)(9,10,13,17,14,19,28,34)(11,20,23,37,22,35,39,33)(12,16,25,32,26,18,29,36)(15,27,30,21,31,38,40,24);(2,5)(3,8)(6,7)(9,11)(10,15,19,31)(12,21)(13,27)(14,22)(16,20,18,35)(17,33,34,37)(23,25)(24,26)(28,38)(29,39)(30,36,40,32);(9,12,14,26)(10,16,19,18)(11,21,22,24)(13,25,28,29)(15,20,31,35)(17,32,34,36)(23,38,39,27)(30,37,40,33)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=i,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, 57409);