#10042592

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

4109 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)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(2,6)(3,9)(8,11)(10,12)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(2,6)(3,9)(8,11)(10,12)(18,19)(23,25)(24,26)(30,31)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=i,e^2=g,f^2=h,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, 9949788);