#8873614

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

4079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=hi,e^2=hi,f^2=h,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, 8780810);