#10055591

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

4302 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=i,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, 9962787);