#9390499

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	#55553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297695);