#9007527

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
prime radical	#42067	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18,23,26)(19,24,21,22)(20,25,28,32)(27,31,30,29)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,3,6,9)(2,7,5,11)(4,10,12,13)(8,15,14,16)(17,19,23,21)(18,24,26,22)(20,27,28,30)(25,31,32,29)(34,40)(35,44)(37,47)(42,45);(2,5)(3,9)(4,12)(15,16)(19,21)(20,28)(22,24)(25,32)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,13)(11,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,12)(14,15)(17,21,23,19)(18,22,26,24)(20,27,28,30)(25,31,32,29)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5,6,2)(3,11,9,7)(4,14,12,8)(10,16,13,15)(17,22)(18,19)(20,29)(21,26)(23,24)(25,27)(28,31)(30,32)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,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, 8914723);