#9053987

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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)(3,5)(4,9)(6,8)(7,10)(11,14)(12,13)(15,16)(18,22)(23,28)(24,29)(30,32)(34,40)(35,45)(36,38)(43,48);(1,3,6,10)(2,7,8,5)(4,11,12,16)(9,15,13,14)(20,27)(24,29)(25,31)(30,32)(33,34)(35,46)(36,42)(37,48)(38,44)(39,40)(41,45)(43,47);(2,8)(4,12)(5,7)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,41)(36,37)(38,47)(39,45)(40,46)(42,43)(44,48);(1,4,6,12)(2,9,8,13)(3,11,10,16)(5,14,7,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,39,38)(34,42,40,44)(35,37,45,47)(41,43,46,48);(1,2)(3,7)(4,13)(5,10)(6,8)(9,12)(11,14)(15,16)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,39,47)(34,43,40,48)(35,36,45,38)(41,42,46,44);(1,5,6,7)(2,10,8,3)(4,14,12,15)(9,16,13,11)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,38,39,36)(34,44,40,42)(35,47,45,37)(41,48,46,43)

Polycyclic presentation:

2^9|b^2=gi,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, 8961183);