#8926521

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

4473 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

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, 8833717);