#8445501

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

4547 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)(12,20)(16,23)(19,24)(21,22)(26,31)(28,32)(34,37)(36,39);(3,4)(7,8)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,21,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,6,4)(2,7,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,4,6,3)(2,7,5,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,5,6,2)(3,8,4,7)(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,23)(20,22)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,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, 8352697);