#8429678

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
prime radical	#43963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4636 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,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, 8336874);