#8188209

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	#40840	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4425 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)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,26)(27,33)(28,29)(30,34)(31,32)(35,40)(36,37)(38,39);(1,3)(2,4)(5,6)(7,8)(9,10,14,15)(11,13,17,22)(12,16,21,19)(18,20,23,24)(26,32)(28,33)(34,39)(36,40);(1,4,5,8)(2,6,7,3)(10,15)(13,22)(16,19)(20,24)(25,27,31,29)(26,33,32,28)(30,37,38,35)(34,36,39,40);(1,3)(2,4)(5,6)(7,8)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,29,31,27)(26,28,32,33)(30,37,38,35)(34,36,39,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=i,e^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, 8095405);