#8905167

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

4081 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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,47,44);(2,7)(4,10)(9,13)(12,16)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29)(33,35)(34,39)(36,44)(37,40)(38,43)(41,45)(42,46)(47,48);(2,7)(3,8)(5,14)(12,16)(18,25)(19,20)(22,31)(29,32)(35,43)(36,45)(39,46)(40,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32)(33,36)(34,40)(35,44)(37,39)(38,45)(41,43)(42,48)(46,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,21)(18,20)(19,25)(22,29)(23,24)(26,28)(27,30)(31,32)(33,37)(34,41)(35,40)(36,46)(38,47)(39,45)(42,44)(43,48);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,35)(34,39)(36,44)(37,40)(38,43)(41,45)(42,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812363);