#9716263

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	#49107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,21)(20,29)(22,31)(23,24)(25,30)(26,28)(27,32)(33,34)(35,43)(36,41)(37,47)(38,40)(39,44)(42,45)(46,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,27)(18,21,24,32)(20,28,30,22)(25,31,29,26)(33,35,38,44)(34,39,40,43)(36,45,46,37)(41,47,48,42);(2,7)(6,13)(8,11)(14,15)(18,24)(19,27)(20,30)(26,31)(34,40)(39,43)(41,48)(42,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,29)(26,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,44)(45,47)(46,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,31)(22,29)(23,32)(24,27)(25,28)(26,30)(33,36,38,46)(34,41,40,48)(35,37,44,45)(39,42,43,47);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,22,23,28)(18,26,24,31)(19,20,27,30)(21,25,32,29)(33,37)(34,42)(35,36)(38,45)(39,41)(40,47)(43,48)(44,46)

Polycyclic presentation:

2^9|b^2=h,e^2=g,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, 9623459);