#9863966

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3810 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9771162);