#8749838

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=gh,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, 8657034);