#8749750

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=gh,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, 8656946);