#489892

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
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	#33675	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 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)(19,30)(20,24)(21,29)(22,28)(23,26)(25,33)(27,34)(31,45)(32,44)(35,42)(36,41)(37,40)(38,43)(39,48)(46,47);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(18,24)(20,34)(21,22)(25,39)(26,40)(27,29)(28,41)(30,44)(32,47)(33,38)(35,48)(36,37)(42,45)(43,46);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,19,23,31)(18,25,26,43)(20,35,37,32)(21,38,22,33)(24,39,40,46)(27,44,41,42)(28,45,29,30)(34,48,36,47);(2,9)(8,16)(10,11)(12,15)(18,26)(19,31)(24,40)(27,41)(28,29)(32,35)(33,38)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397088);