#8639908

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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)(3,10,5,7)(4,8,13,16)(11,15,12,14)(17,18,22,23)(19,21,25,27)(20,24,30,32)(26,31,28,29)(33,34,39,44)(35,45,46,40)(36,42,37,41)(38,43,48,47);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(17,19)(18,21)(20,28)(22,25)(23,27)(24,29)(26,30)(31,32)(33,35,39,46)(34,40,44,45)(36,43,37,47)(38,41,48,42);(4,13)(8,16)(11,12)(14,15)(18,23)(19,25)(24,32)(26,28)(33,36,39,37)(34,41,44,42)(35,43,46,47)(38,40,48,45);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,20,22,30)(18,24,23,32)(19,26,25,28)(21,31,27,29)(33,37,39,36)(34,42,44,41)(35,43,46,47)(38,40,48,45);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,16)(10,15)(17,18)(19,27)(20,24)(21,25)(22,23)(26,29)(28,31)(30,32)(33,35)(34,40)(36,47)(37,43)(38,41)(39,46)(42,48)(44,45);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,21,22,27)(18,25,23,19)(20,29,30,31)(24,26,32,28)(33,38,39,48)(34,43,44,47)(35,41,46,42)(36,40,37,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8547104);