#112841

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1122 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,8,10,9,11,7,12)(13,14,18,23,19,22,17,33)(15,24,27,40,28,20,26,36)(16,29,31,21,32,39,34,38)(25,42,30,41,43,37,44,35);(6,10)(7,8)(11,12)(13,15,19,28)(14,20,22,24)(16,30,32,44)(17,27,18,26)(21,37,38,42)(23,40,33,36)(25,34,43,31)(29,35,39,41);(2,4)(6,11)(7,8)(13,16)(14,21)(15,25)(17,34)(18,31)(19,32)(20,35)(22,38)(23,39)(24,41)(26,44)(27,30)(28,43)(29,33)(36,42)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh,g^c=gi

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, 20037);