#8406878

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

3776 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)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(34,36)(39,41)(40,44)(46,47);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(18,23)(19,26)(25,28)(27,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(3,12)(6,15)(8,16)(10,13)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=gh,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, 8314074);