#9015628

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

4073 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=ghi,e^2=i,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, 8922824);