#9481519

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	#37047	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388715);