#8196912

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

4552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104108);