#8919209

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

4052 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)(3,4,8,12)(5,13,14,9)(10,16,15,11)(34,39)(35,43)(41,44)(42,45);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(33,34,38,39)(35,40,43,46)(36,41,47,44)(37,42,48,45);(2,7)(3,8)(5,14)(11,16)(18,23)(19,26)(21,30)(27,29)(35,43)(37,48)(40,46)(42,45);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,35,38,43)(34,40,39,46)(36,45,47,42)(37,44,48,41);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36)(34,41)(35,42)(37,46)(38,47)(39,44)(40,48)(43,45);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37,38,48)(34,42,39,45)(35,44,43,41)(36,46,47,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=hi,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, 8826405);