#8898655

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	#47891	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805851);