#8919410

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826606);