#8209991

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

3874 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8117187);