#10195897

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

4089 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gh,e^2=gh,f^2=hi,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, 10103093);