#8197244

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

4538 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)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,33)(28,36)(29,31)(30,35)(32,37)(34,38)(39,40);(1,3,5,8)(2,6,7,4)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(13,21)(16,20)(18,24)(22,23)(26,31)(30,33)(32,36)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28,29,37)(26,32,31,36)(27,34,35,40)(30,38,33,39);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,26)(27,30)(28,32)(29,31)(33,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=ghi,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, 8104440);