#149803

order	512
nilpotency class	5
isoclinism	isoclinic class of #149789
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#382 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

314 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,e^b=efhi,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, 56999);