#9243567

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
prime radical	#47889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150763);