#8943919

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

4039 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8851115);