#8588881

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

4016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8496077);