#219355

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

484 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)(19,21)(20,27)(22,26)(23,29)(24,30)(25,31)(28,32)(33,34,39,35,38,41,37,43)(36,40,45,42,46,48,47,44);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,21,18)(20,23,28,25)(22,26,30,24)(27,31,32,29)(33,35,39,41,38,43,37,34)(36,44,45,40,46,42,47,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,29,21,28,30,31)(18,23,24,27,19,25,26,32)(33,36,38,46)(34,40,41,48)(35,42,43,44)(37,47,39,45);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(18,19)(23,25)(24,26)(27,32)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,44,48,42)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,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, 126551);