#149647

order	512
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	#391	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dhi,e^b=ef,h^b=hi,e^d=ehi,f^d=fi,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, 56843);