#8092020

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	#45997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4449 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999216);