#8762014

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

4285 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)(5,10)(6,7)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,27,21)(20,25,30,31)(26,32,28,29)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,7)(4,12)(5,9)(11,16)(18,23)(21,24)(25,31)(29,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,7)(3,10)(4,12)(13,15)(17,19,22,27)(18,24,23,21)(20,28,30,26)(25,29,31,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4,6,12)(2,8,7,14)(3,11,10,16)(5,13,9,15)(17,20,22,30)(18,25,23,31)(19,26,27,28)(21,29,24,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,19)(20,29)(22,24)(23,27)(25,26)(28,31)(30,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,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, 8669210);