#8188518

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	#39963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=ghi,e^2=h,f^2=g,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, 8095714);