#8850588

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	#49589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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)(4,12)(5,10)(6,7)(8,13)(11,15)(14,16)(17,18,22,25)(19,21,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,38)(36,44)(37,41)(39,43)(40,46)(42,48)(45,47);(2,7)(4,13)(5,9)(11,14)(17,19,22,27)(18,23,25,21)(20,26,29,32)(24,30,28,31)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(2,7)(3,10)(8,12)(11,14)(20,29)(24,28)(26,32)(30,31)(33,34,37,41)(35,39,43,38)(36,40,46,44)(42,48,45,47);(1,3)(2,5)(4,14)(6,10)(7,9)(8,16)(11,13)(12,15)(17,20,22,29)(18,24,25,28)(19,26,27,32)(21,31,23,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,4)(2,8)(3,11)(5,15)(6,13)(7,12)(9,16)(10,14)(17,21)(18,19)(20,30)(22,23)(24,32)(25,27)(26,28)(29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,5)(2,3)(4,15)(6,9)(7,10)(8,11)(12,14)(13,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8757784);