#9174988

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	#47621	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=gh,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, 9082184);