#8549519

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

4236 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)(4,10)(5,8)(6,7)(11,16)(12,13)(14,15)(35,36)(39,40)(43,44)(46,47);(2,7)(3,10)(8,15)(11,13)(17,18,22,23)(19,24,28,31)(20,25,29,21)(26,27,32,30)(33,34,38,42)(35,39,36,40)(37,41,45,48)(43,46,44,47);(2,7)(4,9)(8,15)(12,16)(18,23)(20,29)(24,31)(26,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)

Polycyclic presentation:

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

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, 8456715);