#8873637

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

4094 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,32,28,25)(33,34)(35,38)(36,44)(37,41)(39,43)(40,46)(42,48)(45,47);(2,7)(5,9)(8,10)(13,16)(17,19,22,30)(18,23,26,27)(20,28,29,21)(24,25,31,32)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(3,11)(8,10)(12,14)(13,16)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,25,27,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,25)(26,31)(27,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,24,27,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31)(33,36,37,46)(34,40,41,44)(35,42,43,45)(38,47,39,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780833);