#8325525

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	#53094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gh,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, 8232721);