#8844490

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

4276 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8751686);