#8963648

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

4227 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=i,e^2=i,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, 8870844);