#8934474

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

4476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841670);