#8739398

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	#41378	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8646594);