#489918

order	512
nilpotency class	4
isoclinism	isoclinic class of #489733
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

660 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)(17,18,23,27)(19,28,31,47)(20,25,22,37)(21,33,36,44)(24,39,41,30)(26,32,45,35)(29,46,38,43)(34,40,48,42);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19,23,31)(18,24,27,41)(20,29,22,38)(21,34,36,48)(25,40,37,42)(26,43,45,46)(28,44,47,33)(30,32,39,35);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,35,36,32)(24,40,41,42)(26,44,45,33)(28,43,47,46)(30,48,39,34);(2,9)(8,16)(10,11)(12,15)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,40,41,42)(26,33,45,44)(28,46,47,43)(30,34,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,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, 397114);