#490023

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61480	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

721 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:

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

Polycyclic presentation:

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

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, 397219);