#9206605

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

4089 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=gh,e^2=hi,f^2=h,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, 9113801);