#8757259

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

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664455);