#8850578

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

4222 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=hi,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, 8757774);