#9206596

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

3944 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)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,28)(31,32)(33,34,39,37)(35,38,42,40)(36,44,46,41)(43,47,48,45);(1,3)(2,6)(4,13)(5,14)(7,10)(8,9)(11,12)(15,16)(17,18,22,25)(19,20,23,26)(21,24,30,29)(27,28,31,32)(34,37)(35,42)(41,44)(43,48);(1,4,7,14)(2,8,11,15)(3,13,10,5)(6,9,12,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35,39,42)(34,40,37,38)(36,43,46,48)(41,47,44,45);(1,4)(2,8)(3,13)(5,10)(6,9)(7,14)(11,15)(12,16)(17,20,22,26)(18,23,25,19)(21,28,30,32)(24,31,29,27)(33,36,39,46)(34,41,37,44)(35,43,42,48)(38,45,40,47);(1,5,7,13)(2,9,11,16)(3,14,10,4)(6,8,12,15)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,37,39,34)(35,38,42,40)(36,44,46,41)(43,45,48,47);(1,6,7,12)(2,10,11,3)(4,9,14,16)(5,15,13,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,35)(36,45)(37,42)(39,40)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113792);