#8179719

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

4523 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,e^2=h,f^2=g,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, 8086915);