#8932537

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

5226 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)(17,18,23,21)(19,27,28,24)(20,31,34,25)(22,35,37,26)(29,38,32,40)(30,39,36,42)(33,41,46,44)(43,48,45,47);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,15,16,12)(19,28)(24,27)(29,32)(30,36)(38,40)(39,42)(43,45)(47,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,32)(21,27)(22,36)(23,28)(25,40)(26,42)(29,34)(30,37)(31,38)(33,43)(35,39)(41,47)(44,48)(45,46);(9,12)(10,11)(13,15)(14,16)(17,20,23,34)(18,25,21,31)(19,29,28,32)(22,33,37,46)(24,38,27,40)(26,41,35,44)(30,43,36,45)(39,47,42,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,21,23,18)(19,27,28,24)(20,25,34,31)(22,35,37,26)(29,38,32,40)(30,42,36,39)(33,41,46,44)(43,47,45,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,22)(18,26)(19,30)(20,33)(21,35)(23,37)(24,39)(25,41)(27,42)(28,36)(29,43)(31,44)(32,45)(34,46)(38,47)(40,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839733);