#7852756

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

4553 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)(9,10,15,20)(11,13,22,16)(12,18,23,17)(14,19,21,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,14)(13,20)(15,22)(17,19)(18,24)(21,23)(25,27,31,38)(26,32,36,37)(28,30,34,39)(29,35,33,40);(9,12,15,23)(10,17,20,18)(11,21,22,14)(13,19,16,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,14)(13,19)(15,23)(16,24)(17,20)(21,22)(25,28,31,34)(26,33,36,29)(27,30,38,39)(32,40,37,35);(1,4,5,2)(3,7,8,6)(9,13,15,16)(10,11,20,22)(12,19,23,24)(14,17,21,18)(25,29,31,33)(26,34,36,28)(27,35,38,40)(30,32,39,37);(9,14)(10,19)(11,12)(13,18)(15,21)(16,17)(20,24)(22,23)(25,30)(26,35)(27,34)(28,38)(29,37)(31,39)(32,33)(36,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7759952);