#8751682

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	C₂×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 8658878);