#107362

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#379
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14558);