#9203867

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	#50480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18,22,20)(19,23,26,24)(21,25,29,28)(27,30,32,31)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(2,7)(4,8)(10,13)(12,14)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(2,7)(3,9)(10,13)(11,16)(19,26)(23,24)(27,32)(30,31)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,3)(2,8)(4,7)(5,11)(6,9)(10,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,20,22,18)(19,24,26,23)(21,25,29,28)(27,30,32,31)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9111063);