#9243410

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	#55010	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9150606);