#8603532

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

4211 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(2,4)(3,10)(8,12)(11,13)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(34,40)(35,36)(42,45)(43,44);(2,4)(7,9)(8,12)(15,16)(20,29)(24,32)(27,28)(30,31)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,32)(26,31)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21,22,19)(18,23,25,26)(20,28,29,27)(24,30,32,31)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=g,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, 8510728);