#8750758

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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)(3,4)(5,6)(7,8)(9,10,15,16)(11,19,13,17)(12,18,22,14)(20,24,23,21)(26,30)(28,35)(31,36)(34,39);(2,6)(3,7)(10,16)(12,22)(17,19)(20,23)(27,33)(31,36)(34,39)(38,40);(1,3,5,7)(2,4,6,8)(9,11,15,13)(10,17,16,19)(12,23,22,20)(14,21,18,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4)(2,7)(3,6)(5,8)(9,12,15,22)(10,18,16,14)(11,20,13,23)(17,21,19,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,13,15,11)(10,19,16,17)(12,23,22,20)(14,21,18,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,23,19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657954);