#8406885

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

3784 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(18,21)(19,27)(20,30)(26,32)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,18,23,21)(19,28,27,24)(20,31,30,25)(22,26,29,32)(34,40)(35,43)(41,42)(44,45);(3,12)(6,15)(8,16)(10,13)(17,19)(18,24)(20,29)(21,28)(22,30)(23,27)(25,32)(26,31)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20,23,30)(18,25,21,31)(19,29,27,22)(24,32,28,26)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,21,23,18)(19,28,27,24)(20,31,30,25)(22,32,29,26)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,22,23,29)(18,26,21,32)(19,30,27,20)(24,31,28,25)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314081);