#489312

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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	#61478	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396508);