#9039423

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	#47911	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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)(3,8,11,15)(4,9,12,16)(5,10,14,13)(19,25)(23,26)(27,31)(30,32)(33,34)(35,40)(36,45)(37,38)(39,42)(41,48)(43,47)(44,46);(2,7)(4,12)(8,15)(10,13)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,39,36)(34,40,42,45)(37,46,41,43)(38,44,48,47);(4,12)(5,14)(9,16)(10,13)(18,22)(20,28)(23,26)(27,31)(33,36,39,35)(34,40,42,45)(37,46,41,43)(38,47,48,44);(1,3)(2,8)(4,13)(5,9)(6,11)(7,15)(10,12)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,4,6,12)(2,9,7,16)(3,10,11,13)(5,8,14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37,39,41)(34,38,42,48)(35,43,36,46)(40,47,45,44);(1,5,6,14)(2,10,7,13)(3,9,11,16)(4,8,12,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,38)(34,41)(35,44)(36,47)(37,42)(39,48)(40,43)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946619);