#8581806

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4275 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,16)(14,15)(18,23)(19,25)(24,30)(26,31)(33,34,39,36)(35,37,40,42)(38,41,46,44)(43,45,47,48);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,15)(14,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(2,7)(3,10)(9,15)(11,16)(20,27)(24,30)(26,31)(28,32)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)

Polycyclic presentation:

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

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, 8489002);