#8750815

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658011);