#10195908

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

4070 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=ghi,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, 10103104);