#9039120

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	#57271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,23,21)(19,20,26,24)(22,29,30,25)(27,31,32,28)(35,38)(40,42)(43,45)(47,48);(1,3,6,13)(2,7,10,11)(4,9,14,12)(5,8,16,15)(18,21)(19,26)(25,29)(27,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(4,14)(5,16)(8,15)(9,12)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,2)(3,7)(4,15)(5,9)(6,10)(8,14)(11,13)(12,16)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,4,6,14)(2,8,10,15)(3,12,13,9)(5,11,16,7)(17,21,23,18)(19,20,26,24)(22,29,30,25)(27,28,32,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,9)(3,8)(4,11)(6,16)(7,14)(10,12)(13,15)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=h,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, 8946316);