#8712695

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

4133 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,40)(36,38)(37,46)(39,41)(42,44)(43,47)(45,48);(2,7)(3,12)(9,14)(10,13)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,39,43)(34,40,41,47)(36,37,45,44)(38,46,48,42);(2,7)(3,12)(9,14)(10,13)(20,29)(24,28)(26,32)(30,31)(34,41)(35,43)(36,45)(42,46);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,36,39,45)(34,38,41,48)(35,44,43,37)(40,42,47,46);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,37,39,44)(34,42,41,46)(35,45,43,36)(38,47,48,40);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,38)(34,36)(35,46)(37,40)(39,48)(41,45)(42,43)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8619891);