#8746009

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4167 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,e^2=gh,f^2=gh,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, 8653205);