#9240951

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	#52938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=gh,f^2=h,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, 9148147);