#8407169

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

3893 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)(19,22)(20,25)(21,26)(23,27)(24,29)(28,32)(30,31)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19,23,29)(18,24,27,22)(20,28,21,31)(25,30,26,32)(35,41)(38,45)(42,47)(46,48);(3,12)(6,15)(8,16)(10,13)(17,20,23,21)(18,25,27,26)(19,28,29,31)(22,32,24,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,21,23,20)(18,26,27,25)(19,28,29,31)(22,32,24,30)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,19)(18,22)(20,28)(21,31)(23,29)(24,27)(25,32)(26,30)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,22)(18,19)(20,30)(21,32)(23,24)(25,31)(26,28)(27,29)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=h,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, 8314365);