#8930523

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

4189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=i,e^2=gh,f^2=g,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, 8837719);