#466606

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=fi,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373802);