#8607496

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

4579 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8514692);