#460785

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

1098 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)(7,9)(8,10)(11,12)(13,14,19,25,20,26,18,24)(15,21,29,40,30,41,28,39)(16,31,34,22,27,43,33,38)(17,35,32,23,37,42,36,44);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,18,28,20,30,19,29)(14,21,24,39,26,41,25,40)(16,32,33,17,27,36,34,37)(22,42,31,23,38,35,43,44);(1,2)(3,4)(6,10)(9,12)(13,16,20,27)(14,22,26,38)(15,17,30,37)(18,34,19,33)(21,23,41,44)(24,43,25,31)(28,32,29,36)(35,39,42,40);(1,3,4,2)(13,17)(14,23)(15,27)(16,30)(18,36)(19,32)(20,37)(21,38)(22,41)(24,35)(25,42)(26,44)(28,34)(29,33)(31,40)(39,43)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,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, 367981);