#9712203

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	#44437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gh,e^2=gh,f^2=hi,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, 9619399);