#7969443

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4490 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)(11,15)(12,22)(13,18)(14,16)(17,19)(20,21)(23,24)(25,26)(27,35)(28,32)(29,38)(30,33)(31,34)(36,40)(37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,23)(14,19)(16,17)(18,24)(21,22)(25,27)(26,30)(28,37)(29,36)(31,33)(32,40)(34,35)(38,39);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,18)(21,23)(25,28,31,29)(26,32,34,38)(27,36,33,37)(30,39,35,40);(1,4,5,3)(2,8,6,7)(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,24)(22,23)(25,29,31,28)(26,32,34,38)(27,36,33,37)(30,40,35,39);(9,12)(10,13)(11,20)(14,18)(15,23)(16,22)(17,21)(19,24)(25,26)(27,30)(28,38)(29,32)(31,34)(33,35)(36,40)(37,39);(9,13)(10,18)(11,21)(12,16)(14,22)(15,20)(17,24)(19,23)(25,30)(26,33)(27,34)(28,39)(29,40)(31,35)(32,37)(36,38)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,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, 7876639);