#8445649

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	#43202	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4381 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=i,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, 8352845);