#8943740

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	#48667	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8850936);