#8745930

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	#55325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 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)(3,7)(4,10)(5,9)(6,8)(11,14)(12,13)(15,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,3,6,10)(2,7,8,4)(5,11,15,13)(9,14,16,12)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(2,8)(3,10)(9,16)(11,13)(20,28)(25,30)(27,29)(31,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,3)(2,4)(5,13)(6,10)(7,8)(9,14)(11,15)(12,16)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,4)(2,3)(5,14)(6,7)(8,10)(9,13)(11,16)(12,15)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,5,6,15)(2,9,8,16)(3,11,10,13)(4,12,7,14)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8653126);