#9915337

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

4407 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)(3,9)(4,8)(5,10)(6,7)(11,12)(13,15)(14,16)(17,18)(19,27)(20,25)(21,30)(22,24)(23,29)(26,31)(28,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,2)(3,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19,22,29)(18,23,24,27)(20,21,28,31)(25,26,32,30)(35,41)(38,45)(42,47)(46,48);(2,7)(3,10)(8,15)(11,16)(18,24)(23,27)(25,32)(26,30)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,13)(2,8,7,15)(3,11,10,16)(5,14,9,12)(17,20)(18,25)(19,21)(22,28)(23,26)(24,32)(27,30)(29,31)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5)(2,3)(4,12)(6,9)(7,10)(8,16)(11,15)(13,14)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,e^2=gh,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, 9822533);