#219309

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,25)(20,22)(21,29)(23,28)(24,27)(26,31)(30,32)(33,34,39,35,38,42,37,41)(36,40,44,43,45,47,46,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(18,22)(19,26)(24,29)(28,32)(33,35,39,42,38,41,37,34)(36,43,44,47,45,48,46,40);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,27,32,20,26,21,28)(18,23,29,31,22,30,24,25)(33,36,38,45)(34,40,42,47)(35,43,41,48)(37,46,39,44);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(18,22)(23,30)(24,29)(25,31)(33,37,38,39)(34,41,42,35)(36,44,45,46)(40,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126505);