#9054446

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	#51567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,18,23,22)(19,26,29,24)(20,21,25,31)(27,32,30,28)(35,38)(40,42)(43,45)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,30)(31,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,20,23,25)(18,21,22,31)(19,27,29,30)(24,28,26,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,21,23,31)(18,25,22,20)(19,28,29,32)(24,30,26,27)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,22,23,18)(19,24,29,26)(20,21,25,31)(27,32,30,28)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,19)(18,26)(20,30)(21,28)(22,24)(23,29)(25,27)(31,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 8961642);