#9720526

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	#53939	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,20)(21,29)(22,23)(24,27)(25,31)(26,30)(28,32)(33,34,38,40)(35,36,42,44)(37,39,47,46)(41,43,45,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(18,23)(20,24)(25,29)(28,30)(33,35)(34,36)(37,45)(38,42)(39,48)(40,44)(41,47)(43,46);(2,7)(3,10)(8,15)(11,12)(17,19,22,27)(18,24,23,20)(21,26,31,32)(25,30,29,28)(33,36,38,44)(34,35,40,42)(37,43,47,48)(39,41,46,45);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20)(18,19)(21,30)(22,24)(23,27)(25,32)(26,29)(28,31)(33,37)(34,39)(35,41)(36,43)(38,47)(40,46)(42,45)(44,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21,22,31)(18,25,23,29)(19,26,27,32)(20,28,24,30)(33,34)(35,36)(37,46)(38,40)(39,47)(41,48)(42,44)(43,45)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,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, 9627722);