#8751678

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

4027 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=gh,e^2=ghi,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, 8658874);