#9053966

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	#51941	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8961162);