#489345

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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	#61491	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)(17,18,24,20)(19,28,32,45)(21,26,23,34)(22,37,40,48)(25,41,29,31)(27,36,35,39)(30,44,33,47)(38,43,46,42);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,25)(20,29)(21,33)(22,38)(23,30)(24,32)(26,43)(27,44)(28,37)(31,39)(34,42)(35,47)(36,41)(40,46)(45,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,20,24,18)(19,29,32,25)(21,26,23,34)(22,27,40,35)(28,41,45,31)(30,42,33,43)(36,48,39,37)(38,44,46,47);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(17,21,24,23)(18,26,20,34)(19,30,32,33)(22,36,40,39)(25,42,29,43)(27,37,35,48)(28,44,45,47)(31,46,41,38)

Polycyclic presentation:

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

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, 396541);