#8889397

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,22,24)(19,23,25,20)(21,29,31,26)(27,28,30,32)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(2,7)(4,13)(5,8)(11,14)(17,19)(18,23)(20,24)(21,30)(22,25)(26,28)(27,31)(29,32)(34,40)(35,44)(37,47)(42,46);(2,7)(3,10)(9,12)(11,14)(18,24)(19,25)(26,29)(27,30)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,19)(18,23)(20,24)(21,27)(22,25)(26,32)(28,29)(30,31)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20,22,23)(18,25,24,19)(21,28,31,32)(26,30,29,27)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,21,22,31)(18,26,24,29)(19,27,25,30)(20,28,23,32)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796593);