#9040478

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

4134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^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, 8947674);