#8926764

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

5238 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8833960);