#8814019

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721215);