#7716345

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
prime radical	#48602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4235 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623541);