#10036828

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

4284 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,39,40)(35,38,36,43)(37,41,45,48)(42,47,44,46);(4,11)(8,14)(10,15)(12,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35)(34,38)(36,39)(37,44)(40,43)(41,46)(42,45)(47,48);(2,7)(3,9)(8,14)(10,15)(18,19)(23,25)(24,26)(30,31)(34,40)(35,36)(37,45)(46,47);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,35)(34,38,40,43)(37,44,45,42)(41,47,48,46);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37)(34,41)(35,42)(36,44)(38,47)(39,45)(40,48)(43,46);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38)(34,35)(36,40)(37,46)(39,43)(41,44)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=i,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, 9944024);