#8424971

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4250 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332167);