#7879724

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

4492 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gh,e^2=h,f^2=gi,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, 7786920);