#9053989

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	#52477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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)(3,7)(4,8)(5,6)(9,11)(10,15)(12,16)(13,14)(18,22)(23,28)(24,29)(30,32)(34,37)(35,41)(40,44)(43,47);(2,6)(7,9)(8,14)(15,16)(20,27)(24,29)(25,31)(30,32)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(1,3,5,11)(2,7,6,9)(4,12,13,10)(8,16,14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,13)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,34,38,37)(35,39,41,42)(36,40,45,44)(43,46,47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,4,5,13)(2,8,6,14)(3,10,11,12)(7,15,9,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,37,38,34)(35,42,41,39)(36,44,45,40)(43,48,47,46)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^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, 8961185);