#8000984

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

4480 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:

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

Polycyclic presentation:

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

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, 7908180);