#8073148

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

4645 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)(9,10,14,16)(11,17,21,15)(12,13,18,23)(19,24,22,20)(25,26,31,33)(27,34,29,32)(28,30,35,40)(36,39,38,37);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,22)(23,24)(25,27,31,29)(26,32,33,34)(28,36,35,38)(30,37,40,39);(1,2,5,4)(3,6,8,7)(10,16)(13,23)(15,17)(20,24)(26,33)(27,29)(30,40)(36,38);(9,11)(10,17)(12,22)(13,20)(14,21)(15,16)(18,19)(23,24)(25,28)(26,30)(27,36)(29,38)(31,35)(32,37)(33,40)(34,39);(1,4,5,2)(3,7,8,6)(9,12,14,18)(10,13,16,23)(11,19,21,22)(15,20,17,24)(25,29,31,27)(26,34,33,32)(28,38,35,36)(30,39,40,37);(9,13)(10,18)(11,20)(12,16)(14,23)(15,22)(17,19)(21,24)(25,30)(26,35)(27,37)(28,33)(29,39)(31,40)(32,38)(34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,e^2=g,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, 7980344);