#7903858

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

4536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7811054);