#9711967

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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, 9619163);