#9729219

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9636415);