#9240963

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=h,f^2=gh,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, 9148159);