#10196006

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

4056 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103202);