#9947397

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

4112 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)(3,6)(4,7)(5,10)(8,14)(9,12)(11,16)(13,15)(17,18)(19,28)(20,21)(22,32)(23,27)(24,30)(25,31)(26,29)(35,41)(38,42)(43,47)(46,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,24)(20,22)(21,26)(23,30)(25,32)(27,28)(29,31)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(2,3)(7,9)(8,10)(15,16)(17,20,23,31)(18,25,27,21)(19,22,30,29)(24,32,28,26)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,20)(18,21)(19,29)(22,30)(23,31)(24,32)(25,27)(26,28)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,20)(19,26)(22,24)(23,25)(27,31)(28,29)(30,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,22,23,29)(18,26,27,32)(19,20,30,31)(21,28,25,24)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9854593);