#9015335

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

4066 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)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(2,6)(7,9)(8,12)(14,15)(17,19,23,31)(18,24,22,30)(20,28,29,21)(25,27,26,32)(34,37)(35,42)(41,45)(43,48);(2,6)(3,10)(4,13)(14,15)(17,20,23,29)(18,25,22,26)(19,21,31,28)(24,32,30,27)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,26)(19,28)(21,31)(22,25)(23,29)(24,32)(27,30)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,28)(18,27,22,32)(19,29,31,20)(24,25,30,26)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,18)(19,30,31,24)(20,25,29,26)(21,32,28,27)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=hi,e^2=i,f^2=h,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, 8922531);