#8436769

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

4380 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343965);