#418395

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1682 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)(3,4)(6,7)(9,10)(11,22)(12,28)(13,32)(14,19)(15,21)(16,20)(17,25)(18,31)(23,24)(26,37)(27,38)(29,39)(30,34)(33,40)(35,36);(5,6)(7,8)(9,11)(10,17)(12,29)(13,33)(14,25)(15,26)(16,27)(18,24)(19,22)(20,37)(21,38)(23,32)(28,35)(30,36)(31,40)(34,39);(2,4)(9,12,14,30)(10,18,19,13)(11,23,25,40)(15,32,16,31)(17,35,22,39)(20,34,21,28)(24,38,33,37)(26,36,27,29);(5,7,8,6)(9,13)(10,12)(11,24)(14,18)(15,28)(16,34)(17,36)(19,30)(20,32)(21,31)(22,29)(23,38)(25,33)(26,39)(27,35)(37,40)

Polycyclic presentation:

2^9|c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 325591);