#149628

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#382	C₂³
socle	C₂²	C₂×#353
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2285 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 356 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)(11,21,24,37)(12,17)(13,28)(14,19,22,34)(15,20)(16,23,32,38)(18,30,36,27)(25,31,26,35)(29,40,39,33);(9,11,22,24,15,26,14,25)(10,16,19,32,20,33,34,40)(12,23,30,38,13,29,27,39)(17,31,18,35,28,37,36,21);(9,12,15,13)(10,17,20,28)(11,23,26,29)(14,27,22,30)(16,31,33,37)(18,34,36,19)(21,32,35,40)(24,38,25,39)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=efhi,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, 56824);