#7969447

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	#37834	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4724 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:

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

Polycyclic presentation:

2^9|b^2=ghi,e^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, 7876643);