#9040597

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	#49989	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

(17,18,23,26)(19,27,24,20)(21,25,31,22)(28,32,30,29)(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)(17,19,23,24)(18,20,26,27)(21,30,31,28)(22,29,25,32)(33,35,40,46)(34,41,43,48)(36,38,37,44)(39,45,47,42);(2,6)(4,13)(7,9)(10,14)(21,31)(22,25)(28,30)(29,32)(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)(17,20,23,27)(18,24,26,19)(21,32,31,29)(22,30,25,28)(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)(18,25)(19,28)(20,29)(22,26)(23,31)(24,30)(27,32)(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)(17,22,23,25)(18,21,26,31)(19,29,24,32)(20,30,27,28)(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=gh,c^2=h,d^2=gh,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, 8947793);