#8437809

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=gh,e^2=h,f^2=h,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, 8345005);