#8841598

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,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, 8748794);