#9040404

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(33,34,37,41)(35,38,44,47)(36,40,45,39)(42,48,43,46);(1,3,6,11)(2,7,5,9)(4,10,12,13)(8,15,14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35,37,44)(34,38,41,47)(36,43,45,42)(39,48,40,46);(2,5)(4,12)(7,9)(10,13)(18,22)(20,29)(23,25)(26,30)(33,36,37,45)(34,39,41,40)(35,42,44,43)(38,46,47,48);(1,3)(2,7)(4,13)(5,9)(6,11)(8,16)(10,12)(14,15)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,35)(34,38)(36,42)(37,44)(39,46)(40,48)(41,47)(43,45);(1,4,6,12)(2,8,5,14)(3,10,11,13)(7,15,9,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,36)(34,40)(35,43)(37,45)(38,46)(39,41)(42,44)(47,48);(1,5,6,2)(3,9,11,7)(4,14,12,8)(10,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,34)(35,38)(36,40)(37,41)(39,45)(42,48)(43,46)(44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,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, 8947600);