#489922

order	512
nilpotency class	4
isoclinism	isoclinic class of #489739
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	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

645 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)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(17,18,23,20)(19,27,30,46)(21,34,38,45)(22,33,36,26)(24,40,28,29)(25,42,32,37)(31,47,39,43)(35,44,48,41);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19,23,30)(18,24,20,28)(21,35,38,48)(22,39,36,31)(25,43,32,47)(26,44,33,41)(27,45,46,34)(29,42,40,37);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,23,18)(19,28,30,24)(21,25,38,32)(22,33,36,26)(27,40,46,29)(31,44,39,41)(34,42,45,37)(35,43,48,47);(2,9)(8,16)(10,11)(12,15)(18,20)(24,28)(25,32)(26,33)(27,46)(34,45)(41,44)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397118);