#491023

order	512
nilpotency class	4
isoclinism	isoclinic class of #491017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33783	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=ei,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,d^b=dg,d^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398219);