#8943993

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=gh,e^2=g,f^2=g,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, 8851189);