#8325465

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	#44975	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4212 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^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, 8232661);