#422077

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

1080 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)(12,18)(13,28)(14,20)(15,21)(16,32)(17,25)(19,30)(23,38)(24,40)(26,35)(27,37)(29,36)(31,34)(33,39);(1,3,4,2)(5,6)(7,8)(9,11)(10,17)(12,24)(13,29)(14,25)(15,31)(16,26)(18,36)(19,38)(20,22)(21,35)(23,27)(28,40)(30,39)(32,34)(33,37);(2,3)(9,12,14,13)(10,18,20,28)(11,23,25,39)(15,30,16,27)(17,33,22,38)(19,32,37,21)(24,26,29,31)(34,40,35,36);(5,7,8,6)(9,13,14,12)(10,19,20,37)(11,24,25,29)(15,30,16,27)(17,33,22,38)(18,32,28,21)(23,26,39,31)(34,36,35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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,d^b=de,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329273);