#9020246

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8927442);