#9910220

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

4413 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)(3,10)(4,9)(5,14)(6,8)(7,12)(11,16)(13,15)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,3)(2,7)(4,11)(5,9)(6,12)(8,10)(13,14)(15,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(2,8)(4,13)(7,10)(11,14)(18,20)(19,26)(25,28)(27,32)(33,35,38,36)(34,40,42,39)(37,43,46,44)(41,48,45,47);(1,4,6,13)(2,9,8,15)(3,11,12,14)(5,10,16,7)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36,38,35)(34,39,42,40)(37,44,46,43)(41,47,45,48);(1,3)(2,10)(4,11)(5,15)(6,12)(7,8)(9,16)(13,14)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,34)(35,40)(36,39)(37,45)(38,42)(41,46)(43,47)(44,48);(1,5,6,16)(2,11,8,14)(3,9,12,15)(4,10,13,7)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817416);