#8399486

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^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, 8306682);