#8398916

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	#44922	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8306112);