#7714744

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	#37190	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4236 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)(12,14)(13,18)(15,16)(19,24)(20,21)(22,23)(25,26)(27,34)(28,37)(29,32)(30,38)(31,33)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,16)(11,12)(18,23)(19,20)(25,27,31,29)(26,32,33,34)(28,35,38,39)(30,36,37,40);(9,11,15,12)(10,17,16,14)(13,20,22,19)(18,21,23,24)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,14,16,17)(13,20,22,19)(18,21,23,24)(25,29,31,27)(26,32,33,34)(28,35,38,39)(30,40,37,36);(9,13,15,22)(10,18,16,23)(11,19,12,20)(14,21,17,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39);(9,14)(10,11)(12,16)(13,21)(15,17)(18,19)(20,23)(22,24)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=g,e^2=i,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, 7621940);