#8750127

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

4257 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,e^2=gh,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, 8657323);