#8750951

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658147);