#457081

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=ei,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, 364277);