#9175018

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	#43963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=gh,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, 9082214);