#8107658

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014854);