#9711961

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	#57004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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)(33,34,39,43)(35,37,45,40)(36,46,48,41)(38,42,44,47);(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)(37,43)(38,48)(39,45)(41,47)(42,46);(2,6)(7,9)(8,12)(15,16)(33,36,39,48)(34,41,43,46)(35,44,45,38)(37,42,40,47);(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)(33,37,39,40)(34,35,43,45)(36,47,48,42)(38,46,44,41);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,38)(34,42)(35,36)(37,46)(39,44)(40,41)(43,47)(45,48);(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)(37,43)(38,48)(39,45)(41,47)(42,46)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=gh,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, 9619157);