#8840340

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=i,f^2=gh,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, 8747536);