#8463235

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	#42448	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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)(10,16)(11,19)(12,21)(18,23)(26,31)(28,36)(32,34)(35,37);(3,4)(7,8)(9,10,15,16)(11,14,19,24)(12,22,21,17)(13,18,20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,20)(13,21)(15,19)(16,24)(17,18)(22,23)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(1,4,6,3)(2,7,5,8)(9,12,15,21)(10,17,16,22)(11,20,19,13)(14,18,24,23)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,13,15,20)(10,18,16,23)(11,21,19,12)(14,17,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,5,6,2)(3,8,4,7)(9,14,15,24)(10,19,16,11)(12,23,21,18)(13,17,20,22)(25,29,30,27)(26,34,31,32)(28,35,36,37)(33,39,38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,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, 8370431);