#10036712

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

4104 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,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, 9943908);