#8445923

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353119);