#8011969

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=ghi,f^2=h,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, 7919165);