#9015327

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

4089 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922523);