#8399494

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	#40046	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,12)(4,6)(5,15)(7,9)(8,16)(10,14)(11,13)(17,18,23,26)(19,22,25,32)(20,21,24,31)(27,28,29,30)(34,39)(40,43)(41,46)(47,48);(1,3,7,16)(2,8,9,12)(4,13,14,5)(6,15,10,11)(17,19)(18,22)(20,29)(21,30)(23,25)(24,27)(26,32)(28,31)(33,34,38,39)(35,40,36,43)(37,41,45,46)(42,47,44,48);(2,9)(6,10)(8,12)(11,15)(19,25)(22,32)(27,29)(28,30)(33,35,38,36)(34,40,39,43)(37,44,45,42)(41,48,46,47);(1,4)(2,10)(3,13)(5,16)(6,9)(7,14)(8,11)(12,15)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,31)(30,32)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(1,5,7,13)(2,11,9,15)(3,14,16,4)(6,8,10,12)(17,21)(18,24)(19,28)(20,26)(22,29)(23,31)(25,30)(27,32)(33,36,38,35)(34,40,39,43)(37,44,45,42)(41,47,46,48);(1,6,7,10)(2,4,9,14)(3,15,16,11)(5,8,13,12)(17,22,23,32)(18,25,26,19)(20,28,24,30)(21,29,31,27)(33,37,38,45)(34,41,39,46)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306690);