#8761951

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669147);