#8993929

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

4198 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)(3,7,11,16)(4,9,13,12)(5,14,15,10)(17,18,22,24)(19,27,28,23)(20,29,30,25)(21,26,31,32)(33,34,37,41)(35,39,43,38)(36,40,45,44)(42,47,48,46);(1,3,6,11)(2,7,8,16)(4,10,13,14)(5,12,15,9)(17,19,22,28)(18,23,24,27)(20,21,30,31)(25,26,29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,8)(3,11)(4,13)(5,15)(18,24)(20,30)(21,31)(23,27)(33,34,37,41)(35,38,43,39)(36,44,45,40)(42,47,48,46);(1,4)(2,9)(3,10)(5,7)(6,13)(8,12)(11,14)(15,16)(17,20,22,30)(18,25,24,29)(19,21,28,31)(23,26,27,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,21)(18,26)(19,20)(22,31)(23,25)(24,32)(27,29)(28,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,15)(2,10,8,14)(3,12,11,9)(4,7,13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,30)(31,32)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=hi,f^2=gh,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, 8901125);