#8135674

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	#36725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 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)(3,8,6,4)(9,10)(11,18)(12,16)(13,23)(14,24)(15,17)(19,20)(21,22)(28,36)(32,39)(34,40)(37,38);(1,3)(2,4)(5,6)(7,8)(11,12)(16,18)(19,21)(20,22)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(9,11,15,12)(10,16,17,18)(13,19,24,21)(14,20,23,22)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,4,5,8)(2,6,7,3)(9,12,15,11)(10,16,17,18)(13,19,24,21)(14,22,23,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4)(2,3)(5,8)(6,7)(9,13)(10,14)(11,19)(12,21)(15,24)(16,20)(17,23)(18,22)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,14)(10,13)(11,20)(12,22)(15,23)(16,19)(17,24)(18,21)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8042870);