#7714516

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	#46826	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=g,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, 7621712);