#10047453

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

4024 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,25)(19,23,27,21)(20,24,28,31)(26,29,32,30)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,27)(18,21,25,23)(20,26,28,32)(24,30,31,29)(35,36)(39,42)(43,44)(47,48);(2,3)(7,9)(8,10)(14,15)(20,28)(24,31)(26,32)(29,30)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,32)(30,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,30,23,29)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21,22,23)(18,19,25,27)(20,29,28,30)(24,32,31,26)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9954649);