#9545890

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	#56665	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3840 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,21,22)(20,25,29,32)(27,31,30,28)(33,34)(35,43)(36,39)(37,41)(38,42)(40,44)(45,47)(46,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,23,21)(18,24,26,22)(20,27,29,30)(25,31,32,28)(33,35)(34,40)(36,43)(37,38)(39,44)(41,46)(42,45)(47,48);(2,6)(3,5)(4,12)(15,16)(19,21)(22,24)(27,30)(28,31)(37,47)(38,48)(41,45)(42,46);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20,23,29)(18,25,26,32)(19,27,21,30)(22,28,24,31)(33,36,39,34)(35,43,44,40)(37,41,47,45)(38,46,48,42);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,21,23,19)(18,22,26,24)(20,27,29,30)(25,31,32,28)(33,37)(34,41)(35,38)(36,45)(39,47)(40,46)(42,43)(44,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22)(18,19)(20,28)(21,26)(23,24)(25,27)(29,31)(30,32)(33,38)(34,42)(35,37)(36,46)(39,48)(40,45)(41,43)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9453086);