#8919223

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

4181 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826419);