#118065

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#399
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

762 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:

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

Polycyclic presentation:

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

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, 25261);