#8011984

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	#50864	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4509 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,30)(28,29)(31,39)(32,34)(33,40)(35,36)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,20,21)(12,23,18,14)(13,24,17,19)(25,27,32,29)(26,28,34,30)(31,35,40,37)(33,36,39,38);(11,22)(14,23)(16,21)(19,24)(25,28,32,30)(26,27,34,29)(31,36,40,38)(33,35,39,37);(9,12)(10,17)(11,23)(13,20)(14,22)(15,18)(16,19)(21,24)(25,29,32,27)(26,28,34,30)(31,37,40,35)(33,36,39,38);(9,13,15,17)(10,18,20,12)(11,24,22,19)(14,21,23,16)(25,30,32,28)(26,27,34,29)(31,38,40,36)(33,35,39,37);(1,3,4,7)(2,6,5,8)(9,14,15,23)(10,19,20,24)(11,12,22,18)(13,21,17,16)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,40)(34,39)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=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, 7919180);