#9716293

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

4029 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)(19,26)(20,25)(21,23)(22,24)(27,32)(28,30)(29,31)(33,34)(35,38)(36,37)(39,40)(41,45)(42,48)(43,46)(44,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,22,21)(18,23,24,26)(20,28,29,27)(25,32,31,30)(33,35,39,42)(34,38,40,48)(36,43,45,47)(37,46,41,44);(2,7)(6,13)(8,11)(14,15)(18,24)(23,26)(25,31)(30,32)(34,40)(35,42)(36,45)(44,46);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)(33,36)(34,41)(35,43)(37,40)(38,44)(39,45)(42,47)(46,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20,22,29)(18,25,24,31)(19,27,21,28)(23,30,26,32)(33,37,39,41)(34,36,40,45)(35,44,42,46)(38,47,48,43);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,21,22,19)(18,26,24,23)(20,27,29,28)(25,30,31,32)(33,38)(34,42)(35,40)(36,44)(37,43)(39,48)(41,47)(45,46)

Polycyclic presentation:

2^9|b^2=h,e^2=gi,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, 9623489);