#8455740

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	#39059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4308 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=h,e^2=ghi,f^2=gh,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, 8362936);