#9040601

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	#50059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947797);