#8180468

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	#59813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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)(6,7)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(27,33)(28,36)(31,34)(32,37);(1,2,5,4)(3,7,8,6)(10,16)(11,12)(18,21)(19,20)(25,26,30,29)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,28,30,36)(26,32,29,37)(27,35,33,39)(31,38,34,40);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,29,30,26)(27,31,33,34)(28,37,36,32)(35,38,39,40)

Polycyclic presentation:

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

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, 8087664);