#8926769

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

5218 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)(11,15)(12,16)(13,14)(18,23)(19,27)(21,33)(25,36)(28,30)(31,45)(38,43)(47,48);(3,7)(5,8)(9,10,13,14)(11,15,16,12)(18,23)(24,35)(25,36)(26,34)(37,39)(38,43)(40,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,24)(20,25)(21,34)(22,23)(26,33)(27,35)(28,37)(29,43)(30,39)(31,46)(32,36)(38,41)(40,45)(42,48)(44,47);(9,12)(10,11)(13,15)(14,16)(17,19,22,27)(18,24,23,35)(20,30,32,28)(21,29,33,41)(25,39,36,37)(26,38,34,43)(31,44,45,42)(40,48,46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,20,22,32)(18,25,23,36)(19,28,27,30)(21,31,33,45)(24,37,35,39)(26,40,34,46)(29,42,41,44)(38,47,43,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,26)(19,29)(20,31)(22,33)(23,34)(24,38)(25,40)(27,41)(28,42)(30,44)(32,45)(35,43)(36,46)(37,47)(39,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=gi,e^2=gi,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, 8833965);