#8075567

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

4635 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:

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

Polycyclic presentation:

2^9|c^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982763);