#10045594

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

4274 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)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(34,38)(36,44)(39,42)(43,45);(2,6)(7,13)(8,14)(15,16)(20,27)(24,30)(26,31)(28,32)(34,38)(35,41)(40,46)(43,45);(3,9)(7,13)(10,11)(15,16)(18,23)(19,25)(24,30)(26,31)(33,34,37,38)(35,42,41,39)(36,40,44,46)(43,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,44)(46,47);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|c^2=i,d^2=hi,e^2=g,f^2=g,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, 9952790);