#8301652

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	#45726	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8208848);