#9713686

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(17,18,24,21)(19,27,23,25)(20,29,22,26)(28,31,30,32)(36,44)(39,46)(43,47)(45,48);(1,3)(2,8)(4,12)(5,14)(6,9)(7,11)(10,15)(13,16)(17,19,24,23)(18,25,21,27)(20,28,22,30)(26,31,29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,4)(3,11)(9,13)(10,14)(17,20,24,22)(18,26,21,29)(19,28,23,30)(25,31,27,32)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,4,7,2)(3,12,11,8)(5,13,15,9)(6,14,16,10)(17,21,24,18)(19,27,23,25)(20,26,22,29)(28,31,30,32)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,5,7,15)(2,9,4,13)(3,10,11,14)(6,8,16,12)(17,22,24,20)(18,26,21,29)(19,28,23,30)(25,32,27,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,6)(2,10)(3,9)(4,14)(5,12)(7,16)(8,15)(11,13)(17,23,24,19)(18,27,21,25)(20,30,22,28)(26,32,29,31)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=h,f^2=g,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, 9620882);