#9717587

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3830 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)(3,8)(4,7)(5,9)(6,14)(10,16)(11,12)(13,15)(18,23)(19,26)(25,31)(27,28)(33,34)(35,41)(36,40)(37,42)(38,43)(39,46)(44,45)(47,48);(1,3)(2,8)(4,12)(5,14)(6,9)(7,11)(10,15)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(2,4)(3,11)(9,13)(10,14)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,29,31,32)(33,36,38,42)(34,40,43,37)(35,44,46,48)(39,47,41,45);(1,4,7,2)(3,12,11,8)(5,13,15,9)(6,14,16,10)(17,20,22,30)(18,25,23,31)(19,27,26,28)(21,29,24,32);(1,5,7,15)(2,9,4,13)(3,10,11,14)(6,8,16,12)(17,21)(18,19)(20,29)(22,24)(23,26)(25,27)(28,31)(30,32)(33,35)(34,41)(36,44)(37,47)(38,46)(39,43)(40,45)(42,48);(1,6)(2,10)(3,9)(4,14)(5,12)(7,16)(8,15)(11,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31)(33,37,38,40)(34,42,43,36)(35,45,46,47)(39,48,41,44)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=i,f^2=ghi,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, 9624783);