#9017459

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

4071 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,22,26)(19,20,24,23)(21,29,31,25)(27,32,30,28);(2,7)(4,15)(8,11)(10,13)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(2,7)(3,12)(4,15)(5,16)(17,19,22,24)(18,23,26,20)(21,30,31,27)(25,28,29,32)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,24)(19,26)(21,28)(22,23)(25,30)(27,29)(31,32)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,31)(18,25,26,29)(19,27,24,30)(20,28,23,32)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=hi,e^2=hi,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, 8924655);