#7714446

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	#47479	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(10,16)(15,17)(18,23)(22,24)(27,29)(32,35)(36,38)(37,39);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,29,31,27)(26,32,34,35)(28,38,33,36)(30,39,40,37);(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621642);