#457172

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,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, 364368);