#489313

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	#61482	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

699 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18,23,28)(19,29,20,25)(21,34,38,47)(22,39,36,27)(24,32,40,30)(26,43,45,37)(31,42,33,44)(35,46,48,41);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19,23,20)(18,24,28,40)(21,35,38,48)(22,33,36,31)(25,34,29,47)(26,44,45,42)(27,46,39,41)(30,43,32,37);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,20,23,19)(18,25,28,29)(21,32,38,30)(22,33,36,31)(24,34,40,47)(26,41,45,46)(27,42,39,44)(35,37,48,43);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(19,20)(24,40)(25,29)(30,32)(31,33)(35,48)(41,46)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,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, 396509);