#456074

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1099 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 363270);