#8458787

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	#40164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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)(3,6)(4,7)(5,8)(9,10)(11,19)(12,17)(13,14)(15,18)(16,23)(20,22)(21,24)(28,36)(32,39)(35,40)(37,38);(1,3,5,7)(2,6,8,4)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(9,12,15,13)(10,17,18,14)(11,20,23,21)(16,24,19,22)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,12)(10,17,18,14)(11,21,23,20)(16,24,19,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,28,30,36)(26,32,34,39)(27,35,33,40)(29,37,31,38);(1,4,5,6)(2,7,8,3)(9,14,15,17)(10,13,18,12)(11,22,23,24)(16,21,19,20)(25,29,30,31)(26,33,34,27)(28,37,36,38)(32,40,39,35)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=hi,f^2=gi,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, 8365983);