#8188060

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

4449 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095256);