#7975018

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

4516 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)(9,10,14,16)(11,15,20,17)(12,13,19,18)(21,23,24,22)(25,26,31,30)(27,35,36,32)(28,39,38,33)(29,34,37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,17)(18,22)(19,24)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,40)(34,39);(1,3,5,4)(2,7,6,8)(10,16)(11,20)(12,19)(22,23)(27,36)(29,37)(32,35)(34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,23)(19,24)(25,28,31,38)(26,33,30,39)(27,37,36,29)(32,40,35,34);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,21)(13,16)(14,19)(15,22)(17,23)(20,24)(25,29,31,37)(26,34,30,40)(27,38,36,28)(32,39,35,33);(1,4,5,3)(2,8,6,7)(9,13,14,18)(10,19,16,12)(11,22,20,23)(15,21,17,24)(25,30,31,26)(27,32,36,35)(28,39,38,33)(29,34,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7882214);