#9711909

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	#56736	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(18,23)(19,25)(26,31)(27,32)(33,34,39,42)(35,43,46,40)(36,37,38,41)(44,48,47,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43);(2,6)(7,9)(8,12)(15,16)(33,36,39,38)(34,41,42,37)(35,44,46,47)(40,48,43,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28)(33,37,39,41)(34,36,42,38)(35,45,46,48)(40,44,43,47);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,38,39,36)(34,41,42,37)(35,44,46,47)(40,45,43,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,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, 9619105);