#8843942

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	#36735	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4102 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:

(1,2,7,10)(3,8,4,9)(5,15,16,6)(11,12,13,14)(17,18,22,23)(19,24,27,21)(20,28,31,25)(26,30,29,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,46)(42,48)(45,47);(1,3,7,4)(2,8,10,9)(5,11,16,13)(6,14,15,12)(18,23)(21,24)(25,28)(30,32)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(3,4)(8,9)(11,13)(12,14)(17,19,22,27)(18,24,23,21)(20,26,31,29)(25,32,28,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,4,7,3)(2,9,10,8)(5,11,16,13)(6,14,15,12)(17,19)(18,21)(20,29)(22,27)(23,24)(25,32)(26,31)(28,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,14)(9,12)(10,15)(17,20,22,31)(18,25,23,28)(19,26,27,29)(21,30,24,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,6)(2,5)(3,12)(4,14)(7,15)(8,13)(9,11)(10,16)(17,21)(18,19)(20,30)(22,24)(23,27)(25,26)(28,29)(31,32)(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=i,c^2=g,d^2=hi,e^2=g,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, 8751138);