#149573

order	512
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#384	C₂³
socle	C₂²	C₂×#353
prime radical	C₂×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2285 ◁ G
LECS	G ▷ #384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 500 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)(11,20,22,15)(12,16)(13,18,27,17)(14,19)(21,34,35,28)(23,29,26,30)(24,32,38,31)(25,33)(36,39,37,40);(9,11)(10,15)(12,21)(13,26)(14,23)(16,28)(17,29)(18,20)(19,30)(22,27)(24,37)(25,36)(31,39)(32,34)(33,40)(35,38);(9,12,14,25)(10,16,19,33)(11,21,23,36)(13,24,27,38)(15,28,30,40)(17,31,18,32)(20,34,29,39)(22,35,26,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56769);