#8761939

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	#45458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669135);